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Step-by-Step Statistics Solutions

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Table and graph, numerical summary, basic probability, discrete distribution, continuous distribution, sampling distribution, confidence interval, hypothesis testing, two population, population variance, goodness of fit, analysis of variance, simple regression, multiple regression, time series analysis.

Statistics 110: Probability

Strategic Practice and Homework Problems

Actively solving practice problems is essential for learning probability. Strategic practice problems are organized by concept, to test and reinforce understanding of that concept.  Homework problems  usually do not say which concepts are involved, and often require combining several concepts. Each of the Strategic Practice documents here contains a set of strategic practice problems, solutions to those problems, a homework assignment, and solutions to the homework assignment. Also included here are the exercises from the  book that are marked with an s, and solutions to those exercises. 

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Browse Course Material

Course info, instructors.

• Dr. Jeremy Orloff
• Dr. Jennifer French Kamrin

Departments

• Mathematics

As Taught In

• Discrete Mathematics
• Probability and Statistics

Learning Resource Types

Introduction to probability and statistics, problem sets with solutions.

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N ( 244 , 15 50 ) N ( 244 , 15 50 )

As the sample size increases, there will be less variability in the mean, so the interval size decreases.

X is the time in minutes it takes to complete the U.S. Census short form. X ¯ X ¯ is the mean time it took a sample of 200 people to complete the U.S. Census short form.

CI: (7.9441, 8.4559)

The level of confidence would decrease, because decreasing n makes the confidence interval wider, so at the same error bound, the confidence level decreases.

• x ¯ x ¯ = 2.2

X ¯ X ¯ is the mean weight of a sample of 20 heads of lettuce.

EBM = 0.07 CI: (2.1264, 2.2736)

The interval is greater, because the level of confidence increased. If the only change made in the analysis is a change in confidence level, then all we are doing is changing how much area is being calculated for the normal distribution. Therefore, a larger confidence level results in larger areas and larger intervals.

The confidence level would increase.

(24.52,36.28)

We are 95 percent confident that the true mean age for winter Foothill College students is between 24.52 and 36.28.

The error bound for the mean would decrease, because as the CL decreases, you need less area under the normal curve (which translates into a smaller interval) to capture the true population mean.

X is the number of hours a patient waits in the emergency room before being called back to be examined. X ¯ X ¯ is the mean wait time of 70 patients in the emergency room.

CI: (1.3808, 1.6192)

• x ¯ x ¯ = 151
• s x s x = 32
• n – 1 = 107

X ¯ X ¯ is the mean number of hours spent watching television per month from a sample of 108 Americans.

CI: (142.92, 159.08)

(2.93, 3.59)

We are 95 percent confident that the true mean number of colors for national flags is between 2.93 colors and 3.59 colors.

The error bound would become EBM = 0.245. This error bound decreases, because as sample sizes increase, variability decreases, and we need less interval length to capture the true mean.

It would decrease, because the z -score would decrease, which would reduce the numerator and lower the number.

X is the number of successes where the woman makes the majority of the purchasing decisions for the household. P ′ is the percentage of households sampled where the woman makes the majority of the purchasing decisions for the household.

CI: (0.5321, 0.6679)

EBM : 0.0679

X is the number of successes where an executive prefers a truck. P ′ is the percentage of executives sampled who prefer a truck.

CI: (0.19432, 0.33068)

EBM : 0.0707

The sampling error means that the true mean can be 2 percent above or below the sample mean.

P ′ is the proportion of voters sampled who said the economy is the most important issue in the upcoming election.

CI: (0.62735, 0.67265);

EBM: 0.02265

the number of girls, ages 8 to 12, in the 5 p.m. Monday night beginning ice-skating class

P ′ ~ N ( 0.8 , ( 0.8 ) ( 0.2 ) 80 ) P ′ ~ N ( 0.8 , ( 0.8 ) ( 0.2 ) 80 )

CI = (0.72171, 0.87829).

(0.72; 0.88)

With 92 percent confidence, we estimate the proportion of girls, ages 8 to 12, in a beginning ice-skating class at the Ice Chalet to be between 72 percent and 88 percent.

The error bound would increase. Assuming all other variables are kept constant, as the confidence level increases, the area under the curve corresponding to the confidence level becomes larger, which creates a wider interval and thus a larger error.

• X is the height of a Swedish male, and is the mean height from a sample of 48 Swedish males.
• Normal. We know the standard deviation for the population, and the sample size is greater than 30.
• CI: (70.151, 71.49)
• EBM = 0.849
• The confidence interval will decrease in size, because the sample size increased. Recall, when all factors remain unchanged, an increase in sample size decreases variability. Thus, we do not need as large an interval to capture the true population mean.
• x ¯ x ¯ = 23.6
• X is the time needed to complete an individual tax form. X ¯ X ¯ is the mean time to complete tax forms from a sample of 100 customers.
• N ( 23.6 , 7 100 ) N ( 23.6 , 7 100 ) because we know sigma.
• (22.228, 24.972)
• EBM = 1.372
• It will need to change the sample size. The firm needs to determine what the confidence level should be and then apply the error bound formula to determine the necessary sample size.
• The confidence level would increase as a result of a larger interval. Smaller sample sizes result in more variability. To capture the true population mean, we need to have a larger interval.
• According to the error bound formula, the firm needs to survey 206 people. Because we increase the confidence level, we need to increase either our error bound or the sample size.
• X is the number of letters a single camper will send home. X ¯ X ¯ is the mean number of letters sent home from a sample of 20 campers.

N 7.9 ( 2.5 20 ) 7.9 ( 2.5 20 )

• CI: (6.98, 8.82)
• The error bound and confidence interval will decrease.
• x ¯ x ¯ = $568,873
• CL = 0.95, α = 1 – 0.95 = 0.05, z α 2 z α 2 = 1.96 EBM = z 0.025 σ n z 0.025 σ n = 1.96 909200 40 909200 40 = $281,764

Alternate solution:

Using the TI-83, 83+, 84, 84+ Calculator

• Press STAT and arrow over to TESTS .
• Arrow down to 7:ZInterval .
• Press ENTER .
• Arrow to Stats and press ENTER .
• σ : 909,200
• x ¯ x ¯ : 568,873
• Arrow down to Calculate and press ENTER .
• The confidence interval is ($287,114, $850,632).
• Notice the small difference between the two solutions—these differences are simply due to rounding error in the hand calculations.
• We estimate with 95 percent confidence that the mean amount of contributions received from all individuals by House candidates is between $287,109 and $850,637.

Use the formula for EBM , solved for n : n =   z 2 σ 2 E B M 2 n =   z 2 σ 2 E B M 2

From the statement of the problem, you know that σ = 2.5, and you need EBM = 1.

z = z 0.035 = 1.812.

(This is the value of z for which the area under the density curve to the right of z is 0.035.)

n =   z 2 σ 2 E B M 2 = 1.812 2 2.5 2 1 2   ≈   20.52 . n =   z 2 σ 2 E B M 2 = 1.812 2 2.5 2 1 2   ≈   20.52 .

You need to measure at least 21 male students to achieve your goal.

• CI: (6244, 11,014)
• It will become smaller.
• x ¯ x ¯ = 2.51
• s x s x = 0.318
• The effective length of time for a tranquilizer
• The mean effective length of time of tranquilizers from a sample of nine patients
• We need to use a Student’s t -distribution, because we do not know the population standard deviation.
• CI: (2.27, 2.76)
• Check student's solution.
• If we were to sample many groups of nine patients, 95 percent of the samples would contain the true population mean length of time.

x ¯ = $ 251 , 854.23 ; x ¯ = $ 251 , 854.23 ;

s =   $ 521 , 130.41 . s =   $ 521 , 130.41 .

Note that we are not given the population standard deviation, only the standard deviation of the sample.

There are 30 measures in the sample, so n = 30, and df = 30 - 1 = 29.

CL = 0.96, so α = 1 - CL = 1 - 0.96 = 0.04.

α 2 = 0.02 t α 2 = t 0.02 α 2 = 0.02 t α 2 = t 0.02 = 2.150.

E B M = t α 2 ( s n ) = 2.150 ( 521 , 130.41 30 )   ~   $ 204 , 561.66 . E B M = t α 2 ( s n ) = 2.150 ( 521 , 130.41 30 )   ~   $ 204 , 561.66 .

x ¯ x ¯ - EBM = $251,854.23 - $204,561.66 = $47,292.57.

x ¯ x ¯ + EBM = $251,854.23 + $204,561.66 = $456,415.89.

We estimate with 96 percent confidence that the mean amount of money raised by all Leadership PACs during the 2011–2012 election cycle lies between $47,292.57 and $456,415.89.

Alternate Solution

The difference between solutions arises from rounding differences.

• X is the number of unoccupied seats on a single flight. X ¯ X ¯ is the mean number of unoccupied seats from a sample of 225 flights.
• We will use a Student’s t-distribution, because we do not know the population standard deviation.
• CI: (11.12 , 12.08)
• CI: (7.64, 9.36)
• The sample should have been increased.
• Answers will vary.
• The sample size would need to be increased, because the critical value increases as the confidence level increases.

X = the number of people who believe that the president is doing an acceptable job;

P ′ = the proportion of people in a sample who believe that the president is doing an acceptable job.

• N ( 0.61 , ( 0.61 ) ( 0.39 ) 1200 ) N ( 0.61 , ( 0.61 ) ( 0.39 ) 1200 )
• CI: (0.59, 0.63)
• Check student’s solution.
• (0.72, 0.82)
• (0.65, 0.76)
• (0.60, 0.72)
• Yes, the intervals (0.72, 0.82) and (0.65, 0.76) overlap, and the intervals (0.65, 0.76) and (0.60, 0.72) overlap.
• We can say that there does not appear to be a significant difference between the proportion of Asian adults who say that their families would welcome a white person into their families and the proportion of Asian adults who say that their families would welcome a Latino person into their families.
• We can say that there is a significant difference between the proportion of Asian adults who say that their families would welcome a white person into their families and the proportion of Asian adults who say that their families would welcome a black person into their families.
• X = the number of adult Americans who believe that crime is the main problem; P′ = the proportion of adult Americans who believe that crime is the main problem.
• Because we are estimating a proportion, that P′ = 0.2 and n = 1,000, the distribution we should use is N ( 0.2 , ( 0.2 ) ( 0.8 ) 1000 ) N ( 0.2 , ( 0.2 ) ( 0.8 ) 1000 ) .
• CI: (0.18, 0.22)
• One way to lower the sampling error is to increase the sample size.
• The stated ± 3 percent represents the maximum error bound. This means that those doing the study are reporting a maximum error of 3 percent. Thus, they estimate the percentage of adult Americans who the percentage of adult Americans who that crime is the main problem to be between 18 percent and 22 percent.
• p′ = (0 .55 + 0 .49) 2 (0 .55 + 0 .49) 2 = 0.52; EBP = 0.55 – 0.52 = 0.03
• No, the confidence interval includes values less than or equal to 0.50. It is possible that less than half of the population believe this.

STAT TESTS A: 1-PropZinterval with x = (0.52)(1,000), n = 1,000, CL = 0.75.

Answer is (0.502, 0.538).

• Yes, this interval does not fall below 0.50, so we can conclude that at least half of all American adults believe that major sports programs corrupt education – but we do so with only 75 percent confidence.

CL = 0.95; α = 1 – 0.95 = 0.05; α 2 α 2 = 0.025; z α 2 z α 2 = 1.96. Use p ′ = q ′ = 0.5.

n =   z α 2 2 p ′ q ′ E B P 2 =   1.96 2 ( 0.5 ) ( 0.5 ) 0.05 2 = 384.16 . n =   z α 2 2 p ′ q ′ E B P 2 =   1.96 2 ( 0.5 ) ( 0.5 ) 0.05 2 = 384.16 .

You need to interview at least 385 students to estimate the proportion to within 5 percent at 95 percent confidence.

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Statistics and probability

Unit 1: analyzing categorical data, unit 2: displaying and comparing quantitative data, unit 3: summarizing quantitative data, unit 4: modeling data distributions, unit 5: exploring bivariate numerical data, unit 6: study design, unit 7: probability, unit 8: counting, permutations, and combinations, unit 9: random variables, unit 10: sampling distributions, unit 11: confidence intervals, unit 12: significance tests (hypothesis testing), unit 13: two-sample inference for the difference between groups, unit 14: inference for categorical data (chi-square tests), unit 15: advanced regression (inference and transforming), unit 16: analysis of variance (anova).

• Math Article

Statistics Questions

Statistics questions based on CBSE syllabus and NCERT guidelines are provided here for students. These questions cover the syllabus of Statistics concept for Class 9, 10 and 11. Practising these problems will help students to score better on this topic. Also, it will help them to participate in competitive exams. Also, find probability questions here at BYJU’S.

Definition : Statistics is a branch of mathematics concerned with collecting meaningful data or information.

Also, read: Statistics

Questions on Statistics with Answers

1. Give any two examples of collecting data from day-to-day life.

A. Increase in population of our country in the last two decades.

B. Number of tables and chairs in a classroom

2. If marks obtained by students in a class test is given as per below:

55 36 95 73 60 42 25 78 75 62

Then arrange the marks from lowest to highest.

Solution: We need to arrange the marks obtained by each student in ascending order:

25 36 42 55 60 62 73 75 78 95

3. Check the following frequency distribution table, consisting of weights of 38 students of a class:

(i) What is class-interval for classes 31 – 35?

(ii) How many students are there in the range of 41-45 kgs?

(i) Class interval = Upper class limit – lower class limit

(ii) For the 41-45 range, there are 14 students.

4. A family with a monthly income of ` 20,000 had planned the following expenditures per month under various heads:

Draw a bar graph for the data above.

5. In a city, the weekly observations made in a study on the cost of the living index are given in the following table:

Draw a frequency polygon for the data above.

Solution: Class- mark = (Upper class limit-lower class limit)/2 = (150 + 140)/2 = 290/2 = 145

Thus we can create a new table with class-mark.

Now with these class marks we can plot the frequency polygon as shown below.

6. Consider a small unit of a factory where there are 5 employees : a supervisor and four labourers. The workers earn a salary of Rs. 5,000 per month each while the supervisor gets Rs. 15,000 per month. Calculate the mean, median and mode of the salaries.

Mean = (5000 + 5000 + 5000 + 5000 + 15000)/5 = 35000/5 = 7000

So, the mean salary is Rs. 7000 per month

To obtain the median, let us arrange the salaries in ascending order:

5000, 5000, 5000, 5000, 15000

Median = (n+1)/2 = (5+1)/2 = 6/2 = 3rd observation

Median = Rs. 5000/-

Mode = Number of times an observation is repeated = Rs.5000/-

7. The distribution in the table below shows the number of wickets taken by bowlers in one-day cricket matches. Find the mean number of wickets using the correct method. What does the mean signify?

Solution: Here, the class size varies, and the class marks (x i ) are large. Apply the step deviation method for a = 200 and h = 20.

So, ʉ = -106/45

x̄ = 200+20(-106/45) = 200 – 47.11 = 152.89

Hence, on an average, the number of wickets taken by these 45 bowlers in one-day cricket is 152.89.

8. A survey conducted on 20 houses in an area by a group of people resulted in the subsequent frequency table for the number of family members in a house:

Find the mode of this data.

Solution: Here the maximum class frequency = 8,

Class corresponding to this frequency = 3 – 5.

So, the modal class = 3 – 5.

Modal class = 3 – 5, lower limit (l) of modal class = 3, class size (h) = 2

Frequency (f 1 ) of the modal class = 8

Frequency (f 0 ) of class preceding the modal class = 7,

Frequency (f 2 ) of class succeeding the modal class = 2.

Now, let us put these values in the formula :

Therefore, the mode of the data above is 3.286.

9. A survey regarding the heights (in cm) of 51 girls of Class X of a school was conducted and the following data were obtained:

Find the median height.

Solution: Observe the below table:

n/2 = 51/2 = 25.5

This observation lies in the class 145 – 150.

l (the lower limit) = 145,

cf (the cumulative frequency of the class preceding 145 – 150) = 11

f (the frequency of the median class 145 – 150) = 18,

h (the class size) = 5

Using the formula,

So, the median height of the girls is 149.03 cm.

Related Articles:

• Important Questions Class 9 Maths Chapter 14 Statistics
• Important Questions Class 10 Maths Chapter 14 Statistics
• Important Questions Class 11 Maths Chapter 15 Statistics

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The Legend of the 'Unsolvable Math Problem'

A student mistook examples of unsolved math problems for a homework assignment and solved them., david mikkelson, published dec. 3, 1996.

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A legend about the "unsolvable math problem" combines one of the ultimate academic wish-fulfillment fantasies — a student not only proves himself the smartest one in his class, but also bests his professor and every other scholar in his field of study — with a "positive thinking" motif that turns up in other urban legends: when people are free to pursue goals unfettered by presumed limitations on what they can accomplish, they just may manage some extraordinary feats through the combined application of native talent and hard work:

A young college student was working hard in an upper-level math course, for fear that he would be unable to pass. On the night before the final, he studied so long that he overslept the morning of the test. When he ran into the classroom several minutes late, he found three equations written on the blackboard. The first two went rather easily, but the third one seemed impossible. He worked frantically on it until — just ten minutes short of the deadline — he found a method that worked, and he finished the problems just as time was called. The student turned in his test paper and left. That evening he received a phone call from his professor. "Do you realize what you did on the test today?" he shouted at the student. "Oh, no," thought the student. I must not have gotten the problems right after all. "You were only supposed to do the first two problems," the professor explained. "That last one was an example of an equation that mathematicians since Einstein have been trying to solve without success. I discussed it with the class before starting the test. And you just solved it!"

And this particular version is all the more interesting for being based on a real-life incident!

One day in 1939, George Bernard Dantzig, a doctoral candidate at the University of California, Berkeley, arrived late for a graduate-level statistics class and found two problems written on the board. Not knowing they were examples of "unsolved" statistics problems, he mistook them for part of a homework assignment, jotted them down, and solved them. (The equations Dantzig tackled are more accurately described not as unsolvable problems, but rather as unproven statistical theorems for which he worked out proofs.)

Six weeks later, Dantzig's statistics professor notified him that he had prepared one of his two "homework" proofs for publication, and Dantzig was given co-author credit on a second paper several years later when another mathematician independently worked out the same solution to the second problem.

George Dantzig recounted his feat in a 1986 interview for the College Mathematics Journal :

It happened because during my first year at Berkeley I arrived late one day at one of [Jerzy] Neyman's classes. On the blackboard there were two problems that I assumed had been assigned for homework. I copied them down. A few days later I apologized to Neyman for taking so long to do the homework — the problems seemed to be a little harder than usual. I asked him if he still wanted it. He told me to throw it on his desk. I did so reluctantly because his desk was covered with such a heap of papers that I feared my homework would be lost there forever. About six weeks later, one Sunday morning about eight o'clock, [my wife] Anne and I were awakened by someone banging on our front door. It was Neyman. He rushed in with papers in hand, all excited: "I've just written an introduction to one of your papers. Read it so I can send it out right away for publication." For a minute I had no idea what he was talking about. To make a long story short, the problems on the blackboard that I had solved thinking they were homework were in fact two famous unsolved problems in statistics. That was the first inkling I had that there was anything special about them. A year later, when I began to worry about a thesis topic, Neyman just shrugged and told me to wrap the two problems in a binder and he would accept them as my thesis. The second of the two problems, however, was not published until after World War II. It happened this way. Around 1950 I received a letter from Abraham Wald enclosing the final galley proofs of a paper of his about to go to press in the Annals of Mathematical Statistics. Someone had just pointed out to him that the main result in his paper was the same as the second "homework" problem solved in my thesis. I wrote back suggesting we publish jointly. He simply inserted my name as coauthor into the galley proof.

Dr. Dantzig also explained how his story passed into the realm of urban legendry:

The other day, as I was taking an early morning walk, I was hailed by Don Knuth as he rode by on his bicycle. He is a colleague at Stanford. He stopped and said, "Hey, George — I was visiting in Indiana recently and heard a sermon about you in church. Do you know that you are an influence on Christians of middle America?" I looked at him, amazed. "After the sermon," he went on, "the minister came over and asked me if I knew a George Dantzig at Stanford, because that was the name of the person his sermon was about." The origin of that minister's sermon can be traced to another Lutheran minister, the Reverend Schuler [sic] of the Crystal Cathedral in Los Angeles. He told me his ideas about thinking positively, and I told him my story about the homework problems and my thesis. A few months later I received a letter from him asking permission to include my story in a book he was writing on the power of positive thinking. Schuler's published version was a bit garbled and exaggerated but essentially correct. The moral of his sermon was this: If I had known that the problem were not homework but were in fact two famous unsolved problems in statistics, I probably would not have thought positively, would have become discouraged, and would never have solved them.

The version of Dantzig's story published by Christian televangelist Robert Schuller contained a good deal of embellishment and misinformation which has since been propagated in urban legend-like forms of the tale such as the one quoted at the head of this page: Schuller converted the mistaken homework assignment into a "final exam" with ten problems (eight of which were real and two of which were "unsolvable"), claimed that "even Einstein was unable to unlock the secrets" of the two extra problems, and erroneously stated that Dantzig's professor was so impressed that he "gave Dantzig a job as his assistant, and Dantzig has been at Stanford ever since."

George Dantzig (himself the son of a mathematician) received a Bachelor's degree from University of Maryland in 1936 and a Master's from the University of Michigan in 1937 before completing his Doctorate (interrupted by World War II) at UC Berkeley in 1946. He later worked for the Air Force, took a position with the RAND Corporation as a research mathematician in 1952, became professor of operations research at Berkeley in 1960, and joined the faculty of Stanford University in 1966, where he taught and published as a professor of operations research until the 1990s. In 1975, Dr. Dantzig was awarded the National Medal of Science by President Gerald Ford.

George Dantzig passed away at his Stanford home at age 90 on 13 May 2005.

Sightings:   This legend is used as the setup of the plot in the 1997 movie Good Will Hunting . As well, one of the early scenes in the 1999 film Rushmore shows the main character daydreaming about solving the impossible question and winning approbation from all.

Albers, Donald J. and Constance Reid.   "An Interview of George B. Dantzig: The Father of Linear Programming."     College Mathematics Journal.   Volume 17, Number 4; 1986   (pp. 293-314).

Brunvand, Jan Harold.   Curses! Broiled Again!     New York: W. W. Norton, 1989.   ISBN 0-393-30711-5   (pp. 278-283).

Dantzig, George B.     "On the Non-Existence of Tests of 'Student's' Hypothesis Having Power Functions Independent of Sigma."     Annals of Mathematical Statistics .   No. 11; 1940   (pp. 186-192).

Dantzig, George B. and Abraham Wald.   "On the Fundamental Lemma of Neyman and Pearson."     Annals of Mathematical Statistics .   No. 22; 1951   (pp. 87-93).

Pearce, Jeremy.   "George B. Dantzig Dies at 90."     The New York Times .   23 May 2005.

By David Mikkelson

David Mikkelson founded the site now known as snopes.com back in 1994.

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BCOC-134: Business Mathematics and Statistics

Ignou solved assignment solution for 2022-23, if you are looking for bcoc-134 ignou solved assignment solution for the subject business mathematics and statistics, you have come to the right place. bcoc-134 solution on this page applies to 2022-23 session students studying in bcomg courses of ignou., looking to download all solved assignment pdfs for your course together, bcoc-134 solved assignment solution by gyaniversity.

Assignment Code: BCOC-134/TMA/2022-23

Course Code: BCOC-134

Assignment Name: Business Mathematics and Statistics

Year: 2022-2023

Verification Status: Verified by Professor

Section – A

Q. 1 Differentiate between Descriptive and Inferential statistics. (10)

Ans ) Difference between Descriptive and Inferential statistics as follows:

Q. 2 Explain functions and importance of Statistics. (10)

Ans ) The Important Functions of Statistics are:

 To Present Facts in a Proper Form: Statisticians make exact generalisations. In India, cotton yields 180 kg per hectare. This is more accurate and persuasive than saying India's cotton yield is poor.

To Simplify Unwieldy and Complex Data: Statistics simplify difficult facts. Data is often incomprehensible. Classifying facts by common traits is necessary to understand them. Take 1,000 industrial workers' weekly pay.

To Provide Techniques for Making Comparison: Statistics helps compare occurrences over time or location. Estimating national income is not a hobby. It's done to compare income over time to see if people's standard of life is rising. In 2005, India's per-capita income rose 10%. This data will illuminate the 2006 Indian standard of living.

To Study Relationship between Different Phenomena: Correlation and regression are used to study variables. Decision-making requires such interactions. For instance, product demand may affect price. Prices usually decrease product demand.

To Forecast Future Values: Some statistical methods predict variable values. A marketing manager can forecast next year's product demand based on 10 years of sales data.

To Measure Uncertainty: Probability theory estimates the likelihood of uncertain events. Probabilities aid decision-making. If you want to estimate your B. Com exam chances, look at the last 10 years' pass rates.

To Test a Hypothesis: Statistical approaches help generate new theories and test hypotheses. A business wants to know if its malaria medicine works. Chi-square Test could help.

To Draw Valid Inferences: Using sample data, statistical approaches can also infer universe (population) features.

To Formulate Policies in Different Fields: Statistics help create social, economic, and corporate policies. For family planning, the government uses essential statistics data. The government gives employees dearness allowance based on consumer price indices.

Importance of Statistics

Ancient statistics was employed only for statecraft. State administrative data included population, births, and deaths. Recently, statistics have expanded to include social and economic aspects. Over time, statistical methods expanded. It currently encompasses almost all sciences—social, physical, and natural. Statistics is used in agriculture, business, sociology, economics, biometry, and more. Thus, statistics is used in most human activities nowadays.

Statistics and State: The State used to keep order. For military and budgetary policymaking, it collected personnel, crime, income, and wealth data. The Welfare State has expanded the state's involvement. Thus, governments worldwide use price, output, consumption, income, and expenditure data to formulate economic and other policies. Planned economic development is used by developing nations like India to improve living conditions. The government must use accurate statistical data to make choices. To create its five-year plans, the government must know the availability of raw materials, capital goods, financial resources, and population distribution by age, sex, income, etc.

Statistics in Economics: Statistical analysis helps solve production, consumption, distribution, and other economic issues. Consumption data may show how different groups of society consume different goods. Economic policies require pricing, wage, consumption, savings, and investment data. Income inequality policies can benefit from national income and wealth data. Engel's Law of Consumption, Income Distribution Law, and others were developed using statistics in economics. Economic planning requires index numbers, time series analysis, regression analysis, etc. Workers receive DA or bonuses based on the consumer price index. Time series analysis can forecast demand. Statistics are increasingly utilised to test economic assumptions.

Statistics in Business and Management: Modern businesses are becoming more complex and demanding as they grow and compete. Professional management has emerged from huge companies' ownership-management separation. Managerial decision-making relies on timely access to statistical data. Thus, statistical data is increasingly employed in business and industry for sales, purchasing, production, marketing, finance, etc. Market and production research, investment policies, product quality control, economic forecasting, auditing, and many more sectors use statistical approaches. Managers must make uncertain decisions in all situations. Statistics can handle such scenarios. Thus, Wallis and Roberts state that “statistics may be seen as a corpus of strategies for making good decisions in the face of uncertainty.”

Q. 3 Discuss various preliminary adjustments that are necessary to do before analysing a time series data. (10)

Ans ) Preliminary Adjustments before Analysing a Time Series Data:

Calendar Variations : The fact that the number of days in each month of the calendar varies slightly from month to month is a fact that is generally known and accepted. Because there are fewer days in February than there are in other months, for instance, the total quantity of goods that are produced during that month may be lower than the quantity that is produced during other months. When we take into account the holidays, we see an even greater degree of variation. As a consequence of this, it is necessary to make adjustments in order to take into account the myriad of calendar differences.

Price Changes : After taking into account the relevant price indices, it is necessary to perform the conversion from monetary values to real values. These real values can then be compared to the monetary values. This is necessary because fluctuations in the general price level are a natural occurrence that cannot be avoided and are therefore unavoidable.

Population changes : There is no question that the rate of population growth is getting faster and faster. As a direct result of this, the data need to be revised in order to take into account the shifts that have taken place in the population. In situations like these, it is possible to calculate per capita values if it is determined that doing so is necessary (dividing original figures by the total population).

Q. 4 Explain with example the properties of matrix multiplication. (10)

Ans ) Associativity : Matrix multiplication is associative. For three matrices A, B and C of order m x n,n x p and p x q respectively,

(AB) C=A (BC)

Distributive over Addition: Matrix Multiplication is distributive over.

Matrix addition: For three matrices A, B and C of order m x n,n x p

and p x q respectively,

A (B+C) = AB+ AC

Identity : For any matrix A of order m x n, there is an identity matrix In

of order n x n and an identity matrix Im of order m x m such that Im A =

Q. 5 Why do we prefer standard deviation to quartile deviation as a measure of dispersion? (10)

Ans ) Dispersion, also known as variability, is a measure of how spread out a set of data is. Standard deviation and quartile deviation are two measures of dispersion that are commonly used in statistical analysis. While both measures provide information about the spread of the data, standard deviation is often preferred over quartile deviation for several reasons.

Firstly, standard deviation is a more precise measure of dispersion than quartile deviation. Standard deviation considers all the data points and their deviations from the mean, whereas quartile deviation only looks at the range of the middle 50% of the data. This means that standard deviation gives a more accurate and detailed measure of the spread of the entire dataset, whereas quartile deviation only considers a portion of the data.

Furthermore, standard deviation is a widely used measure of dispersion, and as such, it is easier to compare and interpret results when standard deviation is used. It is a very common measure of dispersion and is widely used in statistics, probability theory, and other areas of research. This widespread use means that researchers and analysts are more familiar with the interpretation and properties of standard deviation, making it easier to communicate and compare results across studies.

Additionally, standard deviation is mathematically convenient. It has some mathematical properties that make it convenient to use in statistical analysis. For example, it has a simple formula that can be easily calculated, and it can be used in many statistical tests and models. This mathematical convenience makes standard deviation a more versatile and useful measure of dispersion in a wide range of statistical analyses.

Moreover, standard deviation allows for parametric analysis, which is particularly useful in hypothesis testing. Parametric analysis relies on the assumption that the data comes from a normal distribution. Standard deviation is useful in this context because it has the same mathematical properties as the normal distribution. In contrast, quartile deviation is not as useful in this context because it does not have the same mathematical properties as standard deviation.

However, there are situations where quartile deviation may be preferred over standard deviation. For example, if the data is skewed or has extreme outliers, quartile deviation may provide a more robust measure of dispersion. Additionally, quartile deviation may be more appropriate for data that is not normally distributed or has a non-parametric distribution. Quartile deviation may also be preferred if the researcher is interested in the central tendency of the data and not necessarily in the full spread of the data.

In conclusion, while both standard deviation and quartile deviation are measures of dispersion, standard deviation is often preferred over quartile deviation for several reasons. Standard deviation is a more precise measure of dispersion, it is widely used, mathematically convenient, and allows for parametric analysis. However, in some situations, quartile deviation may be more appropriate, such as when the data is skewed or has extreme outliers, or when the researcher is interested in the central tendency of the data. Ultimately, the choice of measure of dispersion will depend on the research question, the type of data, and the statistical analysis being performed.

Section – B

Q. 6 What are the rules of differentiation? Explain it with examples. (6)

Ans ) Different rules of differentiation with examples.

Constant Function Rule : The derivative of a constant function is always zero. That is, if f(x) = c (where c is a constant), then f'(x) = 0.

If f(x) = 5, then f'(x) = 0.

Power Function Rule: The derivative of a power function is equal to the product of the power and the function raised to the power minus one. That is, if f(x) = xn, then f'(x) = n * x(n-1)

If f(x) = x3, then f'(x) = 3x2.

The Linear Function Rule : The derivative of a linear function is equal to its slope. That is, if f(x) = mx + b, then f'(x) = m.

If f(x) = 2x + 3, then f'(x) = 2.

Rule of Sums and Differences : The derivative of a sum or difference of two or more functions is equal to the sum or difference of their derivatives. That is, if f(x) = g(x) ± h(x), then f'(x) = g'(x) ± h'(x).

If f(x) = x2 + 3x - 1, then f'(x) = 2x + 3.

The Product Rule: The derivative of a product of two functions is equal to the first function times the derivative of the second function plus the second function times the derivative of the first function. That is, if f(x) = g(x) * h(x), then f'(x) = g'(x) * h(x) + g(x) * h'(x).

If f(x) = x2 * sin(x), then f'(x) = 2x * sin(x) + x2 * cos(x).

The Quotient Rule : The derivative of a quotient of two functions is equal to the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator. That is, if f(x) = g(x) / h(x), then f'(x) = (h(x) * g'(x) - g(x) * h'(x)) / h(x)2.

If f(x) = (x2 + 1) / x, then f'(x) = (2x * x - (x2 + 1) * 1) / x^2 = (x2 - 1) / x2.

The Chain Rule : The derivative of a composition of two functions is equal to the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. That is, if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).

If f(x) = sin(x2), then f'(x) = cos(x2) * 2x.

Q. 7 At what rate percent compound interest per annum with Rs. 640 amount to 774.40 in 2 years? (6)

Q. 8 Solve the following equations by using Cramer’s Rule. (6)

Ans) Cramer's Rule is a method for solving a system of linear equations using determinants. To use Cramer's Rule, we need to write the system of equations in the form of:

ax + by = c

dx + ey = f

In this case, we have one equation:

x - 2y = 11

We can rewrite this equation as:

x + (-2y) = 11

So, a = 1, b = -2, and c = 11.

To use Cramer's Rule, we need to find the determinants of the coefficient matrix and the augmented matrix. The coefficient matrix is:

So, both Dx and Dy are equal to 0. This means that the system of equations has either no solution or an infinite number of solutions. In this case, the system of equations is inconsistent and has no solution.

Therefore, the solution using Cramer's Rule is that there is no solution to the given equation.

where Dx is the determinant of the matrix formed by replacing the first column of the coefficient matrix with the augmented column, Dy is the determinant of the matrix formed by replacing the second column of the coefficient matrix with the augmented column, and D is the determinant of the coefficient matrix.

Q. 9 State the characteristics of good measure of variation. (6)

Ans ) A good measure of variation should have the following characteristics:

Simplicity : A good measure of variation should be easy to understand and calculate.

Sensitive : It should be able to detect even small differences in the data.

Applicability : The measure should be applicable to different types of data.

Stability : The measure should not change significantly with the addition or removal of outliers or extreme values.

Relative Independence : The measure should be independent of the unit of measurement used for the data.

Adequacy : The measure should be appropriate for the type of data being analyzed and the purpose of the analysis.

Compatibility : It should be compatible with other measures of central tendency and variability used in the analysis.

Reliability : The measure should produce consistent results when applied to the same data set multiple times.

Robustness : The measure should not be overly affected by extreme values or outliers.

Interpretability : The measure should be easily interpretable and understandable to the intended audience.

Q.10 What do you mean by method of factorization and method of substitution? (6)

Ans ) In mathematics, the method of factorization and method of substitution are two common techniques used to solve equations. The method of factorization involves finding the factors of an equation and then using these factors to simplify the equation into more manageable parts. This method is particularly useful for solving quadratic equations, where the equation is of the form ax^2 + bx + c = 0. To solve such an equation using the method of factorization, we need to factorize the left-hand side of the equation and then set each factor equal to zero. This gives us the values of x that make each factor equal to zero, which are the solutions to the equation.

For example, consider the quadratic equation x2 + 5x + 6 = 0. We can factorize this equation into (x + 3)(x + 2) = 0. Setting each factor equal to zero gives us x = -3 and x = -2 as the solutions to the equation.

The method of substitution involves replacing one or more variables in an equation with other variables or expressions, in order to simplify the equation and make it easier to solve. This method is often used in algebraic equations that involve multiple variables, where we can use one of the variables to eliminate another variable.

For example, consider the system of equations:

2x + 3y = 7

4x - 5y = -1

To solve this system using the method of substitution, we can solve one of the equations for one of the variables in terms of the other variable, and then substitute this expression into the other equation. Let's solve the first equation for y in terms of x:

3y = 7 - 2x

y = (7 - 2x)/3

Now we can substitute this expression for y into the second equation:

4x - 5[(7 - 2x)/3] = -1

We can then simplify and solve for x:

4x - (35/3) + (10x/3) = -1

14x/3 = (32/3)

x = 32/14 = 16/7

Finally, we can substitute this value of x back into one of the equations to solve for y:

2(16/7) + 3y = 7

3y = 7 - (32/7)

Therefore, the solutions to the system of equations are x = 16/7 and y = 3/7

Section – C

Q.11 Write short notes on the following: (5×2)

(a) Regression

Ans ) Regression is a mathematical technique used to analyze the relationship between one or more independent variables and a dependent variable. It involves modeling the relationship between the variables using a mathematical function, which can then be used to predict the value of the dependent variable for any given value of the independent variable(s). The most common type of regression is linear regression, which involves fitting a straight line to the data points. The equation of the line represents the relationship between the independent variable and the dependent variable and can be used to make predictions about the dependent variable for any given value of the independent variable.

Nonlinear regression is another type of regression that is used when the relationship between the variables is not linear. In this case, a more complex mathematical function is used to model the relationship, which can take various forms such as quadratic, exponential, or logarithmic.

Regression analysis involves several steps, including data collection, model specification, parameter estimation, and model validation. Data collection involves gathering the necessary data for the analysis, while model specification involves selecting the appropriate type of regression model and specifying the functional form of the model. Parameter estimation involves estimating the parameters of the model using statistical methods, while model validation involves assessing the goodness of fit of the model and the statistical significance of the estimated parameters.

Regression analysis has numerous applications in many fields of study, including statistics, economics, finance, engineering, social sciences, and healthcare. It is a powerful tool for understanding the relationship between variables and making predictions about the behaviour of a system. It can also be used to identify important factors that influence the dependent variable and to test hypotheses about the relationship between the variables.

(b) Time series

Ans ) Time series is a mathematical method used for analysing and forecasting data that is collected over time. It is a sequence of data points that are collected at regular intervals of time, such as hourly, daily, weekly, or monthly. Time series analysis is used in various fields, including economics, finance, engineering, and social sciences.  The purpose of time series analysis is to identify patterns and trends in the data and to make predictions about future values of the variable. It involves several key steps, including data collection, modeling, and forecasting.

Data collection involves gathering the necessary data for the analysis, while modeling involves selecting the appropriate mathematical model to represent the time series data. There are several types of models used for time series analysis, including autoregressive (AR), moving average (MA), and autoregressive integrated moving average (ARIMA).

Forecasting involves using the time series model to make predictions about future values of the variable. This can be done by extrapolating the pattern in the data, or by using advanced forecasting techniques such as exponential smoothing, trend analysis, or seasonal decomposition.  Time series analysis is used in various applications, such as in the analysis of financial data, weather patterns, and stock prices. It is also used in the analysis of economic trends and consumer behaviour.

Q.12 Differentiate between the following:

(a) Positive correlation and negative correlation

Ans ) The differences between positive and negative correlation:

(b) Geometric mean and harmonic mean

Ans ) The differences between geometric mean and harmonic mean:

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IGNOU ECO-07 July 2023 - January 2024 - Solved Assignment

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This particular Assignment references the syllabus chosen for the subject of Mathematics , for the July 2023 - January 2024 session. The code for the assignment is ECO-07 and it is often used by students who are enrolled in the B.COM, BBA Degree.

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IGNOU ECO-07 (July 2023 - January 2024) Assignment Questions

Q.1 a) Calculate the standard deviation and variance from the following data:

b) What is coefficient of variation? What is its role as a measure of variation?

Q.2 What is sampling? Explain various methods of sampling? Why sample survey is preferred compared to census survey? Explain.

Q.3 Describe the problems that are usually confronted by the investigator when he decides to collect primary data. Explain various methods of collecting primary data and also narrate their merits and demerits.

Q.4 a) A motorist travelled for three days at the rate of 480 km. a day. On the first day he travelled for 10 hours at a speed of 48 km per hour, on the second day he travelled for 12 hours at a speed of 40 km. per hour and on the third day for 15 hours at a speed of 32 km. per hour. What was his average speed? b) Calculate the arithmetic mean for the following data by using the two methods of Arithmetic mean for Discrete Series:

Q.5 Write short notes on the following:

a) Median b) Negative skewness c) Sampling errors d) Objectives of classification

IGNOU ECO-07 (July 2022 - January 2023) Assignment Questions

Q.1 What is sampling? Discuss the various methods of sampling?

Q.2 If the mean of the given frequency distribution is 35, then find the missing frequency y. Also, calculate the median and mode for the distribution.

Q.3 a) Discuss the sources of secondary data. What precautions should be taken while using secondary data? b) Define Geometric and Harmonic mean? Differentiate between them?

Q.4 The wages of nine workers are given below: `310, `290, `320, `280, `300, `290, `320, `310, `280. Find the following: a) Standard Deviation b) Variance

(a) Distrust of Statistics (b) Statistical Derivatives (c) Moving Averages (d) Properties of Normal Curve

1. (क) निम्न समंको से मानक विचलन तथा प्रसरण का परिकलन कीजिये ।

(ख) विचरण गुणांक क्या होता है ? विचरण के एक माप के रप में इसका क्या कार्य है ? यह प्रसरण से किस प्रकार भिन्न है ।

2. प्रतिचयन किसे कहते हैं ? प्रतिचयन की विभिन्न विधियों की व्याख्या कीजिए । संगणना सर्वेक्षण की तुलना में, प्रतिदर्श सर्वेक्षण क्यों अधिमान्य है? व्याख्या कीजिए ।

3. प्राथमिक आंकड़ों के संकलन में किसी अनुसंधानकर्ता को प्राय: जिन समस्याओं का सामना करना पडता है उनका वर्णन कीजिए । प्राथमिक आंकड़ों के संकलन की विभिन्न विधियाँ स्पष्ट कीजिए तथा उनके लाभ और हानियाँ भी बताइए ।

4. (क) एक मोटरगाड़ी वाले ने तीन दिन, 480 कि. मी. प्रतिदिन, यात्रा की। उसने पहले दिन 48 कि. मी. प्रतिघंटे की गति से 10 घंटे यात्रा की, दूसरे दिन 40 कि. मी. प्रतिघंटे की गति से 12 घंटे यात्रा की, और तीसरे दिन 32 कि. मी., प्रतिघंटे की गति से 15 घंटे यात्रा की। उसकी ओउसत गति प्रतिघंटा ज्ञात कीजिए । (ख) असतत श्रेणियों के लिए समांतर माध्य की दोनों विधियों को उपयोग करते हुए निम्न समंको का समांतर माध्य परिकलित कीजिए :

5. निम्नलिखित पर संक्षिप्त टिप्पणीयाँ लिखिए:

(क) मध्यिका इकाई (ख) ऋणात्मक वैषम्य (ग) प्रतिदर्श विभ्रम (घ) वर्गीकरण के उद्देश्य

1. प्रतिचयन क्या है ? प्रतिचयन की विभिन्न विधियों की विवेचना कीजिए ।

2. यदि दिए गए बंटन का माध्य है 35, तो लुप्त आवृत्ति y ज्ञात कीजिए। साथ ही, बंटन के लिए माध्यिका और भूयिष्ठक की गणना कीजिए ।

3. (क) ( द्वितीयक तीयक आँकड़ों के स्रोतों पर चर्चा कीजिए । द्वितीयक आँकड़ों का उपयोग करते समय क्या सावधानियां बरतनी चाहिए ? (ख) गुणोत्तर और हरात्मक माध्य परिभाषित करें? उनके बीच अंतर कीजिए ।

4. नौ श्रमिकों का वेतन नीचे दिया गया है : 310, 290, 320, 280, 300, 290, 320, 310, 280 | निम्नलिखित ज्ञात कीजिए: (क) मानक विचलन (ख) प्रसरण

(क) सांख्यिकी पर अविश्वास (ख) सांख्यिकीय व्युत्पन्न (ग) चल माध्य (घ) सामान्य वक्र की विशेषताएं

ECO-07 Assignment Details

• University IGNOU (Indira Gandhi National Open University)
• Title Elements of Statistics
• Language(s) English and Hindi
• Session July 2023 - January 2024
• Code ECO-07
• Subject Mathematics
• Degree(s) B.COM , BBA
• Course Core Courses (CC)
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BCOC-134 BUSINESS MATHEMATICS AND STATISTICS Solved Assignment 2022-2023

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