Homework 1. Homework 2.

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1 Foundations of Statistics: Homework problems General guideline. While working outside the classroom, use any help you want, including people, computer algebra systems, Internet, and solution manuals, but make sure you are ready for quizzes and and exams, where you are on your own. The experimental project. [The project is due as a hard copy at the lecture on Wednesday, December 4.] Investigate a random shuffle function on your favorite digital music player. Specifically, (1) Based on your experience with the particular shuffle mode [there could be several, so make your pick!], state the appropriate null and alternative hypothesis about the distribution that governs the random shuffle. (2) Design an experiment to test the hypothesis. Keeping in mind the ideas and examples from the book [e.g. the first two chapters] write a short paragraph describing the experiment and explaining why you think your design is reasonable. (3) Conduct the experiment and organize the data you collect. (4) Use a suitable test [e.g. χ-square] to test your hypothesis. (5) Compute the p-value and explain whether the null hypothesis should be rejected or not. (6) Write an overall summary. The final product should be a [reasonable well written and presented] report you wrote yourself, with a reasonably complete understanding of all the details. Both external help and internal collaboration are encouraged. Homework 1. Main topic: Histogram Main objective: Constructing and analyzing a histogram. Problem 1. Given the set of numbers, (a) construct the histogram using identical class intervals of length 5; (b) construct the histogram with 5 class intervals chosen so that each interval contains 5 numbers. Homework 2. Main topics: (a) sample mean, sample standard deviation, etc. (b) normal distribution. Main objectives: (a) computing numerical characteristics of a sample; (b) learning to work with the table of the normal distribution. Problem 1. Given the set of numbers, compute the following: (a) the sample range, (b) the sample mean X, (c) the sample median, (d) the sample standard deviations σ and s, (e) D 1, the average of X X, (f) D 4, the fourth root of the average of (X X) 4. What do you notice about the numbers D 1, σ = D 2, and D 4? 1

2 2 Problem 2. Given a normal random variable X with mean 10 and variance 36, compute the following probabilities: (a) P (X > 5); (b) P (4 < X < 16); (c) P (X < 8); (d) P (X < 20); (e) P (X > 16). Use a table of the standard normal distribution. Problem 3. Compute the variance of the normal random variable X if E(X) = 5 and P (X > 9) = 0.2. Use a table of the standard normal distribution. Homework 3. Main topics: (a) Lines in the plane, (b) correlation coefficient Main objectives: (a) writing and analyzing the equation of the line given various basic information; (b) understanding the basic regression tools. Problem 1. Write the equation of the line (a) passing through the points (1, 2) and ( 5, 2); (b) passing through the point (1, 2) and having slope 5. Problem 2. Given the line y = 2x 5, determine (a) slope (b) intercepts (c) whether the point (1, 1) is on the line. Problem 3. Given the two data sets in Problems 1 of HW1 and HW2, draw the scatter plot, compute the correlation coefficient, and write the equations for the SD line and the regression line. Then order both data sets from smallest to largest and repeat the above computations. Problem 4. Three data sets are collected and the correlation coefficient is computed in each case. The variables are (i) grade point average in freshman year and in sophomore year (ii) grade point average in freshman year and in senior year (iii) length and weight of two-by-four boards Possible values for correlation coefficients are Match the correlations with the data sets; two will be left over. Explain your choices. Problems 5. In a certain class, midterm scores average out to 60 with an SD of 15, as do scores on the final. The correlation between midterm scores and final scores is about The scatter diagram is football-shaped. Predict the final score for a student whose midterm score is (a) 75 (b) 30 (c) 60 (d) unknown Homework 4: review for exam 1 Problem 1. Given the set of numbers 4, 6, 2, 5, 3, 4, 3, 4, 6, 9, 4, 5, 3, 2, 5, 5, 6, 6, 4, 7, 5, 5, 2, 6, 4 compute the following: (a) the sample range, (b) the sample mean X, (c) the sample median, (d) the sample standard deviations σ and s. Then draw the histogram using identical class intervals of length 2. Problem 2. A normal random variable has standard deviation 2 and exceeds 5 with probability 0.3. Compute the average value and the probability to fall below zero. Problem 3. For the first-year students at a certain university, the correlation between SAT scores and first-year GPA was The scatter diagram is football-shaped. Predict the percentile rank on the first-year GPA for a student whose percentile rank on the SAT was (a) 90% (b) 30% (c) 50% (d) unknown Problem 4. Write the equation of the line that passes through the origin and is parallel to the line that passes through the points (0, 2) and ( 1, 0). Homework 5. Main topic: probability

3 Main objective: computing probabilities of various events. Problem 1. Two cards are drawn from a well-shuffled deck. Compute the probability that (a) both are clubs (b) the second is a club given that the first is a club (c) the second is a club given that the first is a diamond. Problem 2. A die is rolled 10 times. Compute the probability of (a) getting 6 ten times (b) not getting 5 ten times (c) getting no more than 4 dots on each roll. Homework 6. Main topic: probability Main objective: computing probabilities of various events. Problem 1. Two dice are rolled. Introduce the following events: (1) E : the sum is odd (2) F : at least one number is 1 (3) G : the sum is 5 List the elementary outcomes in each of the following events: E F, E F, F G, E F, E F G. For this problem, would you care whether the dice are fair? Problem 2. Two fair dice are rolled. Compute the probability that the number on the first is smaller that the number on the second. Problem 3. Let A, B, C be three events such that P (A) = 0.5, P (B) = 0.6, P (C) = 0.8 (a) Can any two of these events be mutually exclusive? Explain your conclusion. (b) Assuming that the events are independent, compute P (A B C). Problem 4. Let A and B be events such that P (A) = 0.7 and P (B) = 0.8. (a) Circle the possible values of P (A B): (b) Circle the possible values of P (A B): You need to explain each of your conclusions. For example, if you think that P (A B) can be 0.5, you draw the corresponding Venn diagram, and if you think that P (A B) cannot be 1, you support your claim with suitable formulas. Problem 5. Of families with 4 children, what proportion have more girls than boys? Problem 6. In each case, you need to decide whether it is better to have 60 rolls or 600 rolls of a fair die: (a) you win 10 dollars if 6 shows at least 20% of the time; (b) you win 10 dollars if 6 shows at least 15% of the time but no more than 20%; (c) you win 10 dollars if 6 shows exactly 16 2 % of the time. 3 Homework 7. Main topic: CLT Main objective: using CLT for approximate computations. Problem 1. A fair die is rolled 600 times. (a) Compute the expected value and standard deviation of the total sum. (b) On each roll, you win 2X 5 dollars, where X is the number of dots the die shows. Compute the expected value and standard deviation of your total winning. Problem 2. A box contains 100 tickets, of which 80 are marked with 10 and the rest, with tickets are drawn with replacement. Let X be the total sum of the numbers on the tickets drawn. (a) Compute µ X. (b) Compute σ X, both directly and using the short-cut from the book. 3

4 4 (c) Use normal approximation to compute the following probabilities: P (X 2600), P (2700 X 3000), P (X < 2900). Problem 3. Let X be the number of cavities that develop in a 6-month period in the mouth of a child that uses the new brand of toothpaste Cavifree. The distribution of X is shown below. c P (X = c) a) A family has three children and they all use Cavifree. Assuming that the number of cavities acquired by any one child is independent of the number acquired by any other child, find the probability that between them they acquire at most one cavity in a 6-month period. b) Find the expected value and the standard deviation of X. c) A boarding school has 150 students and they all use Cavifree. Use the CLT to approximate the probability that the students acquire more than a total of 200 cavities in a 6-month period. (Again, you may assume that the number of cavities acquired by the different students are independent.) Homework 8: review for exam 2. Problem 1. A fair die is rolled twice. What is the probability that one of the numbers is more than twice the other number? Problem 2. A coin is tossed 10 times. Find the chance that there will be exactly 2 heads among the first 5 tosses, and exactly 4 heads among the last 5 tosses. Problem 3. One hundred random draws with replacement are made from the box [1, 6, 7, 9, 9, 10] (a) What are the smallest and largest possible values for the sum of the draws? Are all the values in between possible? (b) The sum is between 650 and 750 with a chance of about (circle one and explain) 1% 10% 50% 90% 99% Problem 4. A large group of people get together. Each one rolls a die 180 times, and counts the number of fives. About what percentage of these people should get counts in the range 15 to 45? Problem 5. A coin is tossed 25 times. Estimate the chance of getting 12 heads and 13 tails. Homework 9. Main topic: estimating proportions Main objective: learning to construct confidence intervals and estimate the required sample size. Problem 1. In a survey of 100 people from a certain city, 20 claimed to have seen a UFO. (a) Find the point estimate of the proportion of people in that city who claim to have seen a UFO. (b) Construct the 98% confidence interval for the proportion of people in that city who claim to have seen a UFO. (c) How many more people in that city must be surveyed to estimate the proportion of the people who claim to have seen a UFO to within ±2% with 98% confidence.

5 Homework 10. Main topic: estimating population average. Main objective: constructing confidence intervals and estimating the required sample size. Problem 1. The lifetime of a toaster from the company Toaster s Choice has a normal distribution with standard deviation 1.5 years. A random sample of 400 toasters was drawn yielding the sample lifetime average of 6 years. a) Compute a 90% confidence interval for the mean lifetime of the toasters. b) What sample size is needed to find the mean lifetime of the toasters to within plus or minus 0.05 years at the same 90% confidence level? c) How will the answers in parts a) and b) change if, instead of knowing the standard deviation to be 1.5 years, it was estimated to be 1.5 years, based on the same sample of 400 devices. d) Do parts a) and b) under the assumption that the lifetime has normal distribution, but with unknown standard deviation, and that a sample of 10 devices produced sample lifetime average of 6 years and sample standard deviation of 1.5 years. e) Compare the intervals from parts a) and d). Which one is longer? Does it make sense? Why? f) Compare the sample sizes in parts b) and d). Which one is larger? Does it make sense? Why? Homework 11. Main topic: Basic hypothesis testing. Main objectives: learning how to use the z test and t test. Problem 1. A weight-loss company Sleek and Slender claims that the average weight loss of its customers is at least 25 pounds. After a bad experience with this company, Fred, an unsatisfied customer, wants to perform sampling in order to reject the claim of Sleek and Slender and possibly sue them. He decides to take 4 randomly chosen customers and to use a level of significance of 5%. He has obtained the data representing their weight loss in pounds. He assumes that the weight-loss data for customers are normally distributed. a) Formulate an appropriate null and alternative hypotheses for Fred to use. b) State the rejection rule. c) Compute the value of the test statistic. d) Should Fred reject the null hypothesis at the 5% level of significance? Explain. e) What can you say about the p-value for for this experiment. Circle one, and explain. (1) The p-value is less than (2) The p-value is between 0.01 and (3) The p-value is between and (4) The p-value is between 0.05 and 0.1. (5) The p-value is greater than 0.1. Problem 2. After losing a game with friends, Alice suspects that the die which was used was not fair. She suspects that the probability of 1 appearing is not 1/6 and she decides to test this by rolling the die 300 times and using a level of significance of 5%. a) Formulate an appropriate null and alternative hypotheses for Alice to use. b) State the rejection rule. c) After rolling the die 300 times, she noticed that 1 appeared 38 times. Compute the value of the test statistic in part b). d) Should Alice reject the null hypothesis at the 5% level of significance? Explain. 5

6 6 e) Compute the p-value. Homework 12. Main topics: two-sample z-test and χ 2 test. Main objective: to understand how they work. Problem 1. In 1970, 59% of college freshmen thought that capital punishment should be abolished; by 2005, the percentage had dropped to 35%. Is the difference real, or can it be explained by chance? You may assume that the percentages are based on two independent simple random samples, each of size 1,000. Problem 2. A study reports that freshmen at public universities work 10.2 hours a week for pay, on average, and the SD is 8.5 hours; at private universities, the average is 8.1 hours and the SD is 6.9 hours. Assume these data are based on two independent simple random samples, each of size 1,000. Is the difference between the averages due to chance? If not, what else might explain it? Problem 3. As part of a study on the selection of grand juries in Alameda county, the educational level of grand jurors was compared with the county distribution: Educational level County Number of jurors Elementary 28.4% 1 Secondary 48.5% 10 Some college 11.9% 16 College degree 11.2% 35 Total 100.0% 62 Could a simple random sample of 62 people from the county show a distribution of educational level so different from the county-wide one? Choose one option and explain. (i) This is absolutely impossible. (ii) This is possible, but fantastically unlikely. (iii) This is possible but unlikely-the chance is around 1 % or so. (iv) This is quite possible-the chance is around 10% or so. (v) This is nearly certain. Problem 4. In a certain town, there are about one million eligible voters. A simple random sample of size 10,000 was chosen, to study the relationship between sex and participation in the last election. The results: Men Women Voted 2,792 3,591 Didn t vote 1,486 2,131 Make a χ 2 -test of the null hypothesis that sex and voting are independent. Homework 13: final review, part 1. Problem 1. Suppose men always married women who were exactly (a) 8% shorter, (b) 2 inches higher, (c) (c) (5% minus 2 inches) shorter. What would the correlation between their heights be in each case? Problem 2. A die is rolled 6 times. Find the chance of getting 3 aces (ones) and 3 sixes. Problem 3. A box contains 3 red tickets and 2 green ones. Five draws will be made at random. You win $1 if 3 of the draws are red and 2 are green. Would you prefer the draws to be made with or without replacement? Why? Problem 4. A certain town has 25,000 families. These families own 1.6 cars, on the average; the SD is And 10% of them have no cars at all. As part of an opinion survey, a

7 simple random sample of 1,500 families is chosen. What is the chance that between 9% and 11% of the sample families will not own cars? Show work. Problem 5. The following results were obtained for about 1,000 families: average height of husband 68 inches, SD 2.7 inches; average height of wife 63 inches, SD 2.5 inches, r = (a) What percentage of the women were over 5 feet 8 inches? (b) Of the women who were married to men of height 6 feet, what percentage were over 5 feet 8 inches? [One foot is 12 inches] Homework 14: final review, part 2. Problem 1. A market researcher for a consumer electronics company wants to study the television viewing habits of residents of a particular city. A random sample of N respondents is selected, and each respondent is instructed to keep a detailed record of all television viewing in a particular week. For this sample the viewing time per week has a mean of 15.3 hours and a sample standard deviation of 3.8 hours. Assume that the amount of time of television viewing per week is normally distributed. Construct a 95% confidence interval for the mean amount of television watched per week in this city for (a) N = 11 (b) N = 21 (c) N = 51 (d) N = 111. Problem 2. A study conducted to estimate the proportion of businesses started in the year 2007 that have failed within five years of their startup, revealed that in a random sample of 80 of such businesses 20 of them had failed. Find a point estimate for the proportion p of all businesses that started in 2007 and have failed within 5 years and calculate a 90% confidence interval for p. Problem 3. A university takes a simple random sample of 132 male students and 279 females; 41% of the men and 17% of the women report working more than 10 hours during the survey week. To find out whether the difference in percentages is statistically significant, the investigator starts by computing z = (41 17)/.048. Is anything wrong? What is the right way to do it and what is the conclusion? Problem 4. Here are the results of 100 rolls of a die: Value No. of times Would you consider the die fair? Explain. Problem 5. In a company of 200 employees, there are 32 employees making at least $100,000 a year. There are 47 employees in the company that have a graduate degree. There are 143 employees that do not have a graduate degree and earn less than $100,000 per year. Based on these number, will you conclude that level of education and salary are dependent? 7