1. True or False. Justify for full credit. (a) A is an event, and Ac is the complement of A, then P(A OR Ac ) = 0. (b) If the variance of a data set is 0, then all the observations in this data set must be identical. (c) If a 95% confidence interval for a population mean contains 1, then the 99% confidence interval for the same parameter must contain 1 (d) When plotted on the same graph, a distribution with a mean of 60 and a standard deviation of 5 will look more spread out than a distribution with a mean of 40 and standard deviation of 8. (e) In a right-tailed test, the value of the test statistic is 2. The test statistic follows a distribution with the distribution curve shown below. If we know the shaded area is 0.03, then we have sufficient evidence to reject the null hypothesis at 0.05 level of significance. 2. Choose the best answer. Justify for full credit. (a) A study was conducted at a local college to analyze the average GPA of students graduated from UMUC in 2015. 100 students graduated from UMUC in 2015 were randomly selected, and the average GPA for the group is 3.5. The value 3.5 is a (i) statistic (ii) parameter (iii) cannot be determined (b) The hotel ratings are usually on a scale from 0 star to 5 stars. The level of this measurement is (i) interval (ii) nominal (iii) ordinal (iv) ratio (c) In a career readiness research, 100 students were randomly selected from the psychology program, 150 students were randomly selected from the communications program, and 120 students were randomly selected from cyber security program. This type of sampling is called: (i) cluster STAT 200: Introduction to Statistics Final Examination, Fall 2016 OL1/US1 Page 3 of 8 (ii) convenience (iii) systematic (iv) stratified 3. Choose the best answer. Justify for full credit. (a) A study of 10 different weight loss programs involved 500 subjects. Each of the 10 programs had 50 subjects in it. The subjects were followed for 12 months. Weight change for each subject was recorded. You want to test the claim that the mean weight loss is the same for the 10 programs. What statistical approach should be used? (i) t-test (ii) linear regression (iii) ANOVA (iv) confidence interval (b) A STAT 200 instructor teaches two classes. She wants to test if the variances of the score distribution for the two classes are different. What type of hypothesis test should she use? (i) t-test for two independent samples (ii) t-test for matched samples (iii) z-test for two samples (iv) F- test 4. The frequency distribution below shows the distribution for IQ scores for a random sample of 1000 adults. (Show all work. Just the answer, without supporting work, will receive no credit.) IQ Scores IQ Scores Frequency Relative Frequency 50 - 69 23 70 - 89 249 90 -109 0.450 110 - 129 130 - 149 25 Total 1000 (a) Complete the frequency table with frequency and relative frequency. Express the relative frequency to three decimal places. (b) What percentage of the adults in this sample has an IQ score of at least 110? (c) Does this distribution have positive skew or negative skew? Why or why not? The five-number summary below shows the grade distribution of a STAT 200 quiz for a sample of 500 students. Answer each question based on the given information, and explain your answer in each case. (a) What is the interquartile range in the grade distribution? (b) Which score band has the most students? (i) 20 - 60 (ii) 60 - 85 (iii) 80 - 100 (Iv) Cannot be determined (c) How many students are in the score band between 60 and 70? 6. Consider selecting one card at a time from a 52-card deck. What is the probability that the first card is an ace and the second card is also an ace? (Note: There are 4 aces in a deck of cards) (Show all work. Just the answer, without supporting work, will receive no credit.) (a) Assuming the card selection is without replacement. (b) Assuming the card selection is with replacement. 7. There are 1000 students in a high school. Among the 1000 students, 350 students take AP Statistics, and 300 students take AP French. 100 students take both AP courses. Let S be the event that a randomly selected student takes AP Statistics, and F be the event that a randomly selected student takes AP French. Show all work. Just the answer, without supporting work, will receive no credit. (a) Provide a written description of the complement event of (S OR F). (b) What is the probability of complement event of (S OR F)? STAT 200: Introduction to Statistics Final Examination, fall 2016 OL1/US1 8. Consider rolling a fair 6-faced die twice. Let A be the event that the sum of the two rolls is at least 10, and B be the event that the first one is a multiple of 3. (a) What is the probability that the sum of the two rolls is at least 10 given that the first one is a multiple of 3? Show all work. Just the answer, without supporting work, will receive no credit. (b) Are event A and event B independent? Explain. 9. Answer the following two questions. (Show all work. Just the answer, without supporting work, will receive no credit). (a) The steering committee of UMUC Green Solutions Team consists of 3 committee members. 15 people are interested in serving in the committee. How many different ways can the committee be selected? (c) A bike courier needs to make deliveries at 6 different locations. How many different routes can he take? 10. Assume random variable x follows a probability distribution shown in the table below. Determine the mean and standard deviation of x. Show all work. Just the answer, without supporting work, will receive no credit. x -2 -1 0 1 2 P(x) 0.1 0.1 0.3 0.2 0.3 11. Mimi joined UMUC basketball team since summer 2016. On average, she is able to score 30% of the field goals. Assume she tries 15 field goals in a game. (a) Let X be the number of field goals that Mimi scores in the game. As we know, the distribution of X is a binomial probability distribution. What is the number of trials (n), probability of successes (p) and probability of failures (q), respectively? (b) Find the probability that Mimi scores at least 2 of the 15 field goals. (round the answer to 3 decimal places) Show all work. Just the answer, without supporting work, will receive no credit. 12. A research concludes that the number of hours of exercise per week for adults is normally distributed with a mean of 3.5 hours and a standard deviation of 3 hours. Show all work. Just the answer, without supporting work, will receive no credit. (a) Find the 80th percentile for the distribution of exercise time per week. (Round the answer to 2 decimal places) (b) What is the probability that a randomly selected adult has more than 7 hours of exercise per week? (Round the answer to 4 decimal places) 13. Assume the SAT Mathematics Level 2 test scores are normally distributed with a mean of 500 and a standard deviation of 100. Show all work. Just the answer, without supporting work, will receive no credit. (a) Consider all random samples of 64 test scores. What is the standard deviation of the sample means? (b) What is the probability that 64 randomly selected test scores will have a mean test score that is between 475 and 525? 14. An insurance company checks police records on 600 randomly selected auto accidents and notes that teenagers were at the wheel in 90 of them. Construct a 95% confidence interval estimate of the proportion of auto accidents that involve teenage drivers. Show all work. Just the answer, without supporting work, will receive no credit. 15. A city built a new parking garage in a business district. For a random sample of 64 days, daily fees collected averaged $2,000, with a standard deviation of $400. Construct a 95% confidence interval estimate of the mean daily income this parking garage generates. Show all work. Just the answer, without supporting work, will receive no credit. 16. ABC Company claims that the proportion of its employees investing in individual investment accounts is higher than national proportion of 45%. A survey of 100 employees in ABC Company indicated that 50 of them have invested in an individual investment account. Assume Mimi wants to use a 0.10 significance level to test the claim. (a) Identify the null hypothesis and the alternative hypothesis. (b) Determine the test statistic. Show all work; writing the correct test statistic, without supporting work, will receive no credit. (c) Determine the P-value for this test. Show all work; writing the correct P-value, without supporting work, will receive no credit. (d) Is there sufficient evidence to support ABC Company's claim that the proportion of its employees investing in individual investment accounts is higher than 45%? Explain. 17. In a study of memory recall, 5 people were given 10 minutes to memorize a list of 20 words. Each was asked to list as many of the words as he or she could remember both 1 hour and 24 hours later. The result is shown in the following table. Number Of Words Recalled Subject 1 2 3 4 5 1 Hour 14 18 11 13 12 24 Hours Later 12 15 9 12 12 18. In a pulse rate research, a simple random sample of 40 men results in a mean of 80 beats per minute, and a standard deviation of 11.3 beats per minute. Based on the sample results, the researcher concludes that the pulse rates of men have a standard deviation greater than 10 beats per minutes. Use a 0.05 significance level to test the researcher's claim. (a) Identify the null hypothesis and alternative hypothesis. (b) Determine the test statistic. Show all work; writing the correct test statistic, without supporting work, will receive no credit. (c) Determine the P-value for this test. Show all work; writing the correct P-value, without supporting work, will receive no credit. (d) Is there sufficient evidence to support the researcher's claim? Explain. 19 Answer: (a) The null and alternative hypotheses are, H 0 : The five types of candy bags are equally popular, 1 0.20 5 H a : The five types of candy bags are not equally popular. i.e. p1 p2 p3 p4 p5 (b) The test statistic, 2 = 10.0 (c) The P-value = 0.0404 (d) Critical value, 20.05,4 =9.488 . Since 2 = 10.0 > 9.488, the 2 critical value at .05 level for this righttailed test, reject the null hypothesis. Therefore, there is sufficient evidence to reject the claim that the five types of candy bags are equally popular. Work for (b) and (c): b The expected frequencies, Ei Npi . E1 500 0.20 100 E2 500 0.20 100 E3 500 0.20 100 E4 500 0.20 100 E5 500 0.20 100 Oi Ei 2 Test Statistic, Ei i 2 125 100 2 85 100 2 105 100 2 90 100 2 100 100 100 100 2 95 100 100 10.0 c 20 p -value P 2 4 10.0 0.0404 Answer: (a) The required regression line is, ^y = 0.421 x + 43.270 (b) The required predicted value at x = 80 is, 81.17. Work for (a) and (b): a From the given data, xi 445, yi 447, xi 2 34625, yi 2 33959, x y i i 33835 and n 6 The least square line of the form y a bx to the given data be fitted as follows, Slope b n xi yi xi yi n xi xi 2 2 6 33835 445 447 6 34625 445 2 0.421 1 1 yi b xi 447 0.421 445 43.270 n 6 The required least square line is, y 0.421 x 43.270 Intercept a b The predicted final exam score if the average quiz score is 90 y90 0.421 90 43.270 81.17 Answer: (a) The null and alternative hypotheses are, H 0 : The five types of candy bags are equally popular, 1 0.20 5 H a : The five types of candy bags are not equally popular. i.e. p1 p2 p3 p4 p5 (b) The test statistic, 2 = 10.0 (c) The P-value = 0.0404 19 2 (d) Critical value, 0.05,4 = 9.488. Since 2 = 10.0 > 9.488, the 2 critical value at .05 level for this right-tailed test, reject the null hypothesis. Therefore, there is sufficient evidence to reject the claim that the five types of candy bags are equally popular. Work for (b) and (c): b The expected frequencies, Ei Npi . E1 500 0.20 100 E2 500 0.20 100 E3 500 0.20 100 E4 500 0.20 100 E5 500 0.20 100 Oi Ei 2 Test Statistic, Ei i 2 125 100 2 85 100 2 105 100 2 90 100 2 100 100 100 100 2 95 100 100 10.0 c p -value P 2 4 10.0 0.0404 Answer: (a) The required regression line is, = 0.421 x + 43.270 (b) The required predicted value at x = 80 is, 81.17. Work for (a) and (b): 20 a From the given data, xi 445, yi 447, xi 2 34625, yi 2 33959, x y i i 33835 and n 6 The least square line of the form y a bx to the given data be fitted as follows, Slope b n xi yi xi yi n xi 2 xi 2 6 33835 445 447 6 34625 445 2 0.421 1 1 yi b xi 447 0.421 445 43.270 n 6 The required least square line is, y 0.421 x 43.270 Intercept a b The predicted final exam score if the average quiz score is 90 y90 0.421 90 43.270 81.17 19 Answer: (a) The null and alternative hypotheses are, H 0 : The five types of candy bags are equally popular, 1 0.20 5 H a : The five types of candy bags are not equally popular. i.e. p1 p2 p3 p4 p5 (b) The test statistic, 2 = 10.0 (c) The P-value = 0.0404 (d) Critical value, 20.05,4 =9.488 . Since 2 = 10.0 > 9.488, the 2 critical value at .05 level for this righttailed test, reject the null hypothesis. Therefore, there is sufficient evidence to reject the claim that the five types of candy bags are equally popular. Work for (b) and (c): b The expected frequencies, Ei Npi . E1 500 0.20 100 E2 500 0.20 100 E3 500 0.20 100 E4 500 0.20 100 E5 500 0.20 100 Oi Ei 2 Test Statistic, Ei i 2 125 100 2 85 100 2 105 100 2 90 100 2 100 100 100 100 2 95 100 100 10.0 c 20 p -value P 2 4 10.0 0.0404 Answer: (a) The required regression line is, ^y = 0.421 x + 43.270 (b) The required predicted value at x = 80 is, 81.17. Work for (a) and (b): a From the given data, xi 445, yi 447, xi 2 34625, yi 2 33959, x y i i 33835 and n 6 The least square line of the form y a bx to the given data be fitted as follows, Slope b n xi yi xi yi n xi xi 2 2 6 33835 445 447 6 34625 445 2 0.421 1 1 yi b xi 447 0.421 445 43.270 n 6 The required least square line is, y 0.421 x 43.270 Intercept a b The predicted final exam score if the average quiz score is 90 y90 0.421 90 43.270 81.17 Answer: (a) The null and alternative hypotheses are, H 0 : The five types of candy bags are equally popular, 1 0.20 5 H a : The five types of candy bags are not equally popular. i.e. p1 p2 p3 p4 p5 (b) The test statistic, 2 = 10.0 (c) The P-value = 0.0404 19 2 (d) Critical value, 0.05,4 = 9.488. Since 2 = 10.0 > 9.488, the 2 critical value at .05 level for this right-tailed test, reject the null hypothesis. Therefore, there is sufficient evidence to reject the claim that the five types of candy bags are equally popular. Work for (b) and (c): b The expected frequencies, Ei Npi . E1 500 0.20 100 E2 500 0.20 100 E3 500 0.20 100 E4 500 0.20 100 E5 500 0.20 100 Oi Ei 2 Test Statistic, Ei i 2 125 100 2 85 100 2 105 100 2 90 100 2 100 100 100 100 2 95 100 100 10.0 c p -value P 2 4 10.0 0.0404 Answer: (a) The required regression line is, = 0.421 x + 43.270 (b) The required predicted value at x = 80 is, 81.17. Work for (a) and (b): 20 a From the given data, xi 445, yi 447, xi 2 34625, yi 2 33959, x y i i 33835 and n 6 The least square line of the form y a bx to the given data be fitted as follows, Slope b n xi yi xi yi n xi 2 xi 2 6 33835 445 447 6 34625 445 2 0.421 1 1 yi b xi 447 0.421 445 43.270 n 6 The required least square line is, y 0.421 x 43.270 Intercept a b The predicted final exam score if the average quiz score is 90 y90 0.421 90 43.270 81.17