STAT 200: Introduction to Statistics Final Examination, Spring 2016 OL1/US1 Page 1 of 7 STAT 200 OL1/US1 Sections Final Exam Spring 2016 This is an open-book exam. You may refer to your text and other course materials as you work on the exam, and you may use a calculator. You must complete the exam individually. Neither collaboration nor consultation with others is allowed. It is a violation of the UMUC Academic Dishonesty and Plagiarism policy to use unauthorized materials or work from others. Answer all 20 questions. Make sure your answers are as complete as possible. Show all of your work and reasoning. In particular, when there are calculations involved, you must show how you come up with your answers with critical work and/or necessary tables. Answers that come straight from calculators, programs or software packages will not be accepted. If you need to use software (for example, Excel) and /or online or hand-held calculators to aid in your calculation, you must cite the sources and explain how you get the results. Test has 20 questions. Questions 15,16,17,18,19 will be graded as 2 points, the rest of questions - 1 point. Total 25 points available for the Final Exam. You must include the Honor Pledge on the title page of your submitted final exam. Exams submitted without the Honor Pledge will not be accepted. STAT 200: Introduction to Statistics 1. Final Examination, Spring 2016 OL1/US1 Page 2 of 7 True or False. Justify for full credit. (a) (b) (c) (d) (e) The standard deviation of a data set cannot be negative. If P(A) = 0.4 , P(B) = 0.5, and A and B are disjoint, then P(A AND B) = 0.2. The mean is always equal to the median for a normal distribution. A 95% confidence interval is wider than a 98% confidence interval of the same parameter. In a two-tailed test, the value of the test statistic is 1.5. If we know the test statistic follows a Student's t-distribution with P(T < 1.5) = 0.98, then we fail to reject the null hypothesis at 0.05 level of significance . 2. Identify which of these types of sampling is used: cluster, convenience, simple random, systematic, or stratified. Justify for full credit. (a) A STAT 200 professor wants to estimate the study hours of his students. He teaches two sections, and plans on randomly selecting 10 students from the first section and 15 students from the second section. A STAT 200 student is interested in the number of credit cards owned by college students. She surveyed all of her classmates to collect sample data. The quality control department of a semiconductor manufacturing company tests every 100th product from the assembly line. On the day of the last presidential election, UMUC News Club organized an exit poll in which specific polling stations were randomly selected and all voters were surveyed as they left those polling stations. (b) (c) (d) 3. The frequency distribution below shows the distribution for checkout time (in minutes) in UMUC MiniMart between 3:00 and 4:00 PM on a Friday afternoon. (Show all work. Just the answer, without supporting work, will receive no credit.) Checkout Time (in minutes) Frequency Relative Frequency 1.0 - 1.9 3 2.0 - 2.9 12 0.20 3.0 - 3.9 4.0 - 4.9 3 5.0 -5.9 Total (a) (b) (c) 25 Complete the frequency table with frequency and relative frequency. Express the relative frequency to two decimal places. What percentage of the checkout times was at least 4 minutes? Does this distribution have positive skew or negative skew? Why? STAT 200: Introduction to Statistics Final Examination, Spring 2016 OL1/US1 Page 3 of 7 4. A box contains 3 marbles, 1 red, 1 green, and 1 blue. Consider an experiment that consists of taking 1 marble from the box, then replacing it in the box and drawing a second marble from the box. (Show all work. Just the answer, without supporting work, will receive no credit.) (a) (b) List all outcomes in the sample space. What is the probability that at least one marble is red? fraction form) 5. The five-number summary below shows the grade distribution of two STAT 200 quizzes for a sample of 500 students. (Express the answer in simplest Minimum Quiz 1 Quiz 2 Q1 Median Q3 Maximum 15 20 30 35 55 50 85 90 100 100 For each question, give your answer as one of the following: (i) Quiz 1; (ii) Quiz 2; (iii) Both quizzes have the same value requested; (iv) It is impossible to tell using only the given information. Then explain your answer in each case. (a) (b) (c) Which quiz has less interquartile range in grade distribution? Which quiz has the greater percentage of students with grades 90 and over? Which quiz has a greater percentage of students with grades less than 50? 6. There are 1000 students in a high school. Among the 1000 students, 800 students have a laptop, and 300 students have a tablet. 250 students have both devices. Let L be the event that a randomly selected student has a laptop, and T be the event that a randomly selected student has a tablet. Show all work. Just the answer, without supporting work, will receive no credit. (a) (b) Provide a written description of the event L OR T. What is the probability of event L OR T? 7. Consider rolling two fair dice. Let A be the event that the two dice land on different numbers, and B be the event that the first one lands on 6. (a) What is the probability that the first one lands on 6 given that the two dice land on different numbers? Show all work. Just the answer, without supporting work, will receive no credit. Are event A and event B independent? Explain. (b) STAT 200: Introduction to Statistics 8. (a) (b) 9. Final Examination, Spring 2016 OL1/US1 Page 4 of 7 There are 8 books in the \"Statistics is Fun\" series. (Show all work. Just the answer, without supporting work, will receive no credit). How many different ways can Mimi arrange the 8 books in her book shelf? Mimi plans on bringing two of the eight books with her in a road trip. How many different ways can the two books be selected? 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 P(x) -2 0.1 0 0.2 1 0.3 3 0.1 5 0.3 10. Mimi just started her tennis class three weeks ago. On average, she is able to return 20% of her opponent's serves. Assume her opponent serves 10 times. (a) Let X be the number of returns that Mimi gets. 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? Find the probability that that she returns at least 1 of the 10 serves from her opponent. Show all work. Just the answer, without supporting work, will receive no credit. (b) 11. Assume the weights of men are normally distributed with a mean of 172 lb and a standard deviation of 30 lb. Show all work. Just the answer, without supporting work, will receive no credit. (a) (b) Find the 80th percentile for the distribution of men's weights. What is the probability that a randomly selected man is greater than 185 lb? 12. Assume the IQ scores of adults are normally distributed with a mean of 100 and a standard deviation of 15. Show all work. Just the answer, without supporting work, will receive no credit. (a) (b) If a random sample of 25 adults is selected, what is the standard deviation of the sample mean? What is the probability that 25 randomly selected adults will have a mean IQ score that is between 95 and 105? 13. A survey showed that 80% of the 1600 adult respondents believe in global warming. Construct a 95% confidence interval estimate of the proportion of adults believing in global warming. Show all work. Just the answer, without supporting work, will receive no credit. STAT 200: Introduction to Statistics Final Examination, Spring 2016 OL1/US1 Page 5 of 7 14. In a study designed to test the effectiveness of acupuncture for treating migraine, 100 patients were randomly selected and treated with acupuncture. After one-month treatment, the number of migraine attacks for the group had a mean of 2 and standard deviation of 1.5. Construct a 95% confidence interval estimate of the mean number of migraine attacks for people treated with acupuncture. Show all work. Just the answer, without supporting work, will receive no credit. 15. Mimi is interested in testing the claim that more than 75% of the adults believe in global warming. She conducted a survey on a random sample of 100 adults. The survey showed that 80 adults in the sample believe in global warming. Assume Mimi wants to use a 0.05 significance level to test the claim. (a) (b) (c) (d) 16. Identify the null hypothesis and the alternative hypothesis. Determine the test statistic. Show all work; writing the correct test statistic, without supporting work, will receive no credit. Determine the P-value for this test. Show all work; writing the correct P-value, without supporting work, will receive no credit. Is there sufficient evidence to support the claim that more than 75% of the adults believe in global warming? Explain. 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. Subject 1 2 3 4 5 Number of Words Recalled 1 hour later 24 hours later 14 12 18 15 11 9 13 12 12 12 Is there evidence to suggest that the mean number of words recalled after 1 hour exceeds the mean recall after 24 hours? Assume we want to use a 0.10 significance level to test the claim. (a) (b) (c) (d) Identify the null hypothesis and the alternative hypothesis. Determine the test statistic. Show all work; writing the correct test statistic, without supporting work, will receive no credit. Determine the P-value for this test. Show all work; writing the correct P-value, without supporting work, will receive no credit. Is there sufficient evidence to support the claim that the mean number of words recalled after 1 hour exceeds the mean recall after 24 hours? Justify your conclusion. STAT 200: Introduction to Statistics 17. (c) (d) 18. Brown 42 Yellow 21 (b) 19. Orange 12 Green 7 Tan 18 Identify the null hypothesis and the alternative hypothesis. Determine the test statistic. Show all work; writing the correct test statistic, without supporting work, will receive no credit. Determine the P-value. Show all work; writing the correct P-value, without supporting work, will receive no credit. Is there sufficient evidence to support the claim that the published color distribution is correct? Justify your answer. A random sample of 4 professional athletes produced the following data where x is the number of endorsements the player has and y is the amount of money made (in millions of dollars). x y (a) Page 6 of 7 The UMUC Daily News reported that the color distribution for plain M&M's was: 40% brown, 20% yellow, 20% orange, 10% green, and 10% tan. Each piece of candy in a random sample of 100 plain M&M's was classified according to color, and the results are listed below. Use a 0.05 significance level to test the claim that the published color distribution is correct. Show all work and justify your answer. Color Number (a) (b) Final Examination, Spring 2016 OL1/US1 0 1 1 2 2 4 5 8 Find an equation of the least squares regression line. Show all work; writing the correct equation, without supporting work, will receive no credit. Based on the equation from part (a), what is the predicted value of y if x = 3? Show all work and justify your answer. A farmer is interested in whether there is any variation in the weights of apples between two trees. Data collected from the two trees are as follows: Her null hypothesis and alternative hypothesis are: (a) Determine the test statistic. Show all work; writing the correct test statistic, without supporting work, will receive no credit. STAT 200: Introduction to Statistics (b) (c) Final Examination, Spring 2016 OL1/US1 Page 7 of 7 Determine the P-value for this test. Show all work; writing the correct P-value, without supporting work, will receive no credit. Is there sufficient evidence to justify the rejection of H 0 at the significance level of 0.05? Explain. 20. A study of 5 different weight loss programs involved 250 subjects. Each program was followed by 50 subjects for 12 months. Weight change for each subject was recorded. Mimi wants to test the claim that the mean weight loss is the same for the 5 programs. (a) Complete the following ANOVA table with sum of squares, degrees of freedom, and mean square (Show all work): Source of Variation Factor (Between) Error (Within) Total (b) (c) (d) Sum of Squares (SS) Degrees of Freedom (df) Mean Square (MS) 42.36 1100.76 249 Determine the test statistic. Show all work; writing the correct test statistic, without supporting work, will receive no credit. Determine the P-value for this test. Show all work; writing the correct P-value, without supporting work, will receive no credit. Is there sufficient evidence to support the claim that the mean weight loss is the same for the 5 programs at the significance level of 0.01? Explain. Answer all 25 questions. Make sure your answers are as complete as possible. Show all of your work and reasoning. In particular, when there are calculations involved, you must show how you come up with your answers with critical work and/or necessary tables. Answers that come straight from programs or software packages will not be accepted. If you need to use software (for example, Excel) and /or online or hand-held calculators to aid in your calculation, please cite the sources and explain how you get the results. 1. True or False. Justify for full credit. (a) If the variance of a data set is zero, then all the observations in this data set are zero. No, all the observations will be same but it is not necessary that all the observations will be zero, because if all the observations are same even then the variance will be zero. (b) If P(A) = 0.4 , P(B) = 0.5, and A and B are disjoint, then P(A AND B) = 0.9. No, because if A and B are disjoint then P(A AND B) = 0. (c) Assume X follows a continuous distribution which is symmetric about 0. If , then . Yes, P(X<-3) will be less than 0.3 because the value will be towards negative side. (d) A 95% confidence interval is wider than a 90% confidence interval of the same parameter. Yes, a 95% confidence interval is wider than a 90% confidence interval of the same parameter because increasing the confidence limit we increases the acceptance region and reduces the rejection region. (e) In a right-tailed test, the value of the test statistic is 1.5. If we know the test statistic follows a Student's t-distribution with P(T < 1.5) = 0.96, then we fail to reject the null hypothesis at 0.05 level of significance . No, we will not be able to reject the null hypothesis as we reject the null hypothesis is test statistic is greater than the tabulated value of t. Refer to the following frequency distribution for Questions 2, 3, 4, and 5. Show all work. Just the answer, without supporting work, will receive no credit. The frequency distribution below shows the distribution for checkout time (in minutes) in UMUC MiniMart between 3:00 and 4:00 PM on a Friday afternoon. Checkout minutes) Time (in Frequency 1.0 - 1.9 3 2.0 - 2.9 Relative Frequency 12 3.0 - 3.9 4.0 - 4.9 0.20 3 5.0 -5.9 Total 2. 25 Complete the frequency table with frequency and relative frequency. Express the relative frequency to two decimal places. Checkout Time (in minutes) Frequency Relative Frequency 1.0 - 1.9 0.12 2.0 - 2.9 12 0.48 3.0 - 3.9 5 0.20 4.0 - 4.9 3 0.12 5.0 -5.9 2 0.08 Total 3. 3 25 1 What percentage of the checkout times was at least 3 minutes? Percentage of the checkout times was at least 3 minutes = (5 + 3 +2)/25*100 = 40% 4. In what class interval must the median lie? Explain your answer. We will make the intervals continuous by subtracting 0.05 from lower limit and adding 0.05 in upper limit therefore the corresponding intervals will be 0.95-1.95, 1.95 - 2.95, 2.95 - 3.95, 3.95 4.95, 4.95 -5.95. N=25 therefore, N/2=12.5, the immediate less observation lies in 2.0-2.9 i.e. 1.95-2.95. 5. Does this distribution have positive skew or negative skew? Why? The distribution is positively skewed as the frequency in the second interval is 12 and all the proceeding frequencies are very less. This can also be seen in the following graph Frequency 14 12 10 8 Frequency Frequency 6 4 2 0 1.0 - 1.9 2.0 - 2.9 3.0 - 3.9 4.0 - 4.9 5.0 -5.9 Checkout T ime (in minutes) Refer to the following information for Questions 6 and 7. Show all work. Just the answer, without supporting work, will receive no credit. Consider selecting one card at a time from a 52-card deck. (Note: There are 4 aces in a deck of cards) 6. If the card selection is without replacement, what is the probability that the first card is an ace and the second card is also an ace? (Express the answer in simplest fraction form) (5 pts) The probability that the first card is an ace and the second card is also an ace: P(First card is an Ace and Second card is also an Ace)=(4/52)*(3/51) =0.004524887 7. If the card selection is with replacement, what is the probability that the first card is an ace and the second card is also an ace? (Express the answer in simplest fraction form) The probability that the first card is an ace and the second card is also an ace: P(First card is an Ace and Second card is also an Ace)=(4/52)*(4/52) = 0.00591716 Refer to the following situation for Questions 8, 9, and 10. The five-number summary below shows the grade distribution of two STAT 200 quizzes for a sample of 500 students. Minimum Q1 Median Q3 Maximum Quiz 1 15 45 55 85 100 Quiz 2 20 35 50 90 100 For each question, give your answer as one of the following: (a) Quiz 1; (b) Quiz 2; (c) Both quizzes have the same value requested; (d) It is impossible to tell using only the given information. Then explain your answer in each case. 8. Which quiz has less interquartile range in grade distribution? Quiz 1 Because the inter quartile range i.e. Q3-Q1=40 for quiz 1 is less than the interquartile range Q3Q1= 55 for quiz 2. 9. Which quiz has the greater percentage of students with grades 90 and over? Quiz 2 Because 75th percentile of quiz 2 is having grades 90 and this is not the case for quiz 1. 10. Which quiz has a greater percentage of students with grades less than 60? Quiz 1 Because median of quiz 1 is having grades 55 and this is not the case for quiz 1. Refer to the following information for Questions 11, 12, and 13. Show all work. Just the answer, without supporting work, will receive no credit. There are 1000 students in a high school. Among the 1000 students, 800 students have a laptop, and 300 students have a tablet. 150 students have both devices. P(Laptop)=800/1000, P(tablet)=300/1000, P(Laptop and Tablet)=150/1000 11. What is the probability that a randomly selected student has neither device? The probability that the student has neither laptop nor tablet is: P(Neither Laptop nor Tablet)= 1-P(Laptop or tablet) =1-(800/1000)-(300/1000)+(200/1000) =0.1 12. 13. What is the probability that a randomly selected student has a laptop, given that he/she has a tablet? The probability that a randomly selected student has a laptop, given that he/she has a tablet is: P(student has laptop | student has tablet)= P(Laptop and Tablet)/ P(Tablet) = (150/1000)/(800/1000) =0.1875 Let event A be the selected student having a laptop, and event B be the selected student having a tablet. Are A and B independent events? Why or why not? Yes, the events A and B are independent of each other because the working of laptop and tablets are different and it is the choice of the students to choose any of the one i.e. laptop or tablet or both. 14. A combination lock uses three distinctive numbers between 0 and 49 inclusive. How many different ways can a sequence of three numbers be selected? (Show work) To find the different ways in which a sequence of three numbers can be selected is: n Cr = 50 ! 50 C3 = 3 ! ( 503 ) ! = 19600 15. Let random variable x represent the number of heads when a fair coin is tossed three times. Show all work. Just the answer, without supporting work, will receive no credit. Construct a table describing the probability distribution. Number of Heads (X) 0 1 Probability [P(X=r)] 0.125 0.375 (a) (b) 2 0.375 3 0.125 Determine the mean and standard deviation of x. (Round the answer to two decimal places) Mean= Expected Value of x = x*p(x) = 0*0.125 + 1*0.375 + 2*0.375 + 3*0.125 =1.5 Standard Deviation= sqrt(x*x*p(x)- (x*p(x))^2) =0*0*0.125 + 1*1*0.375 + 2*2*0.375 + 3*3*0.125 -(1.5)^2 =0.866025404 16. Mimi just started her tennis class three weeks ago. On average, she is able to return 20% of her opponent's serves. Assume her opponent serves 10 times. (a) Let X be the number of returns that Mimi gets. 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? n = 10, p = 0.20 and q = 1-0.20 = 0.80 (b) Find the probability that that she returns at least 1 of the 10 serves from her opponent. (Show work) X = number of serves that Mimi returns P(X1) = 1- P(X=0) =1 - 10! ( 0.20 )10 (0.8) 0! ( 100 ) ! =0.99999992 Refer to the following information for Questions 17, 18, and 19. Show all work. Just the answer, without supporting work, will receive no credit. The lengths of mature jalapeo fruits are normally distributed with a mean of 3 inches and a standard deviation of 1 inch. 17. What is the probability that a randomly selected mature jalapeo fruit is between 1.5 and 4 inches long? (5 pts) Here we have = 3, = 1 Z score can be calculated as Z=(X-)/() We need to find P(1.5 1) =0.1587+ 0.1587=0.3174 (c) Is there sufficient evidence to justify the rejection of H0 at the 0.01 level? Explain. No, there sufficient evidence to justify the rejection of H0 at 0.01 level of significance because, as p-value is 0.3174 which is far greater than level of significance. Hence, we will not reject the null hypothesis. 22. Consumption of large amounts of alcohol is known to increase reaction time. To investigate the effects of small amounts of alcohol, reaction time was recorded for five individuals before and after the consumption of 2 ounces of alcohol. Do the data below suggest that consumption of 2 ounces of alcohol increases mean reaction time? Subject 1 2 3 4 5 Reaction Time (seconds) Before After 6 7 8 8 4 6 7 8 9 8 Assume we want to use a 0.01 significance level to test the claim. (a) Identify the null hypothesis and the alternative hypothesis. H0: H1: D =0 D 0 Subject 1 2 3 4 5 Reaction Time (seconds) Before After 6 8 4 7 9 d 7 8 6 8 8 1 0 2 1 -1 0.6 (d-d )^2 0.16 0.36 1.96 0.16 2.56 5.2 (b) Determine the test statistic. Show all work; writing the correct test statistic, without supporting work, will receive no credit. The test statistic is defined as: t= d D /sqrt [ ((di - d )^2 / (n - 1) )/n] - under H0, the test statistic is: t=(0.6-0)/sqrt(5.2/(5-1)/5) =1.176696811 (c) Determine the P-value. Show all work; writing the correct P-value, without supporting work, will receive no credit. P-value: p-value= P(t < -1.176696811) + P(t > 1.176696811) =0.15228061+0.15228061 =0.30456122 (d) Is there sufficient evidence to support the claim that consumption of 2 ounces of alcohol increases mean reaction time? Justify your conclusion. No, there is sufficient evidence to support the claim that consumption of 2 ounces of alcohol increases mean reaction time, because the p-value is greater than the level of significance 0.01 which tells us that we can accept the null hypothesis. 23. The UMUC MiniMart sells four different types of Halloween candy bags. The manager reports that the four types are equally popular. Suppose that a sample of 500 purchases yields observed counts 150, 110, 130, and 110 for types 1, 2, 3, and 4, respectively. Type Number of Bags 1 150 2 110 3 130 4 110 Assume we want to use a 0.10 significance level to test the claim that the four types are equally popular. (a) Identify the null hypothesis and the alternative hypothesis. Here we will use chi-square test for goodness of fit. H0: The data are consistent with the four different types of Halloween candy bags. H1: The data are not consistent with the four different types of Halloween candy bags. (b) Determine the test statistic. Show all work; writing the correct test statistic, without supporting work, will receive no credit. Test statistic. The test statistic is a chi-square random variable ( 2) defined by the following equation. Expected value=(1/4)*500=125 2 = [ (Oi - Ei)2 / Ei ] Where Oi is the observed frequency count for the ith level of the categorical variable, and Ei is the expected frequency count for the ith level of the categorical variable. Type Number of Bags (O) 1 Total 150 2 3 4 E 110 130 110 500 O-E 125 125 125 125 500 (O-E)^2 25.00 -15.00 5.00 -15.00 ((O-E)^2)/E 625 225 25 225 Chi-square test statistics = 8.8 (c) Determine the P-value for the test. Show all work; writing the correct P-value, without supporting work, will receive no credit. With test statistic 8.8 and df 3 p-value lies between 0.05 to 0.025. 5 1.8 0.2 1.8 8.8 (d) Is there sufficient evidence to support the manager's claim that the four types are equally popular? Justify your answer. No, there is no sufficient evidence support that the manager's claim that the four types are equally popular because we are rejecting the null the null hypothesis and accepting the alternative which states that data is not consistent with the four different types of Halloween candy bags 24. A random sample of 4 professional athletes produced the following data where x is the number of endorsements the player has and y is the amount of money made (in millions of dollars). x y (a) 0 1 1 2 3 3 5 8 Find an equation of the least squares regression line. Show all work; writing the correct equation, without supporting work, will receive no credit. The equation of the least squares regression line is: y=b0 + b1*x where b1 is the slope and b0 is the intercept (the point where the line crosses the y axis) y 1 2 3 8 Total b0 = 14 y x 2 x x y 2 n x2 ( x ) n xy x y b1 = x 0 1 3 5 n x ( x ) 2 2 = xy 9 = x^2 0 2 9 40 51 0 1 9 25 35 1435951 31 2 = 59 = 0.525424 435( 9 ) 451914 78 2 = 59 =1.322034 435( 9 ) Therefore, the regression line equation is : y=0.525424 + 1.322034*x (b) Based on the equation from part (a), what is the predicted value of y if x = 4? Show all work and justify your answer. The predicted value of y if x = 4 is y=0.525424 + 1.322034*4 =5.81356 25. A STAT 200 instructor is interested in whether there is any variation in the final exam grades between her two classes Data collected from the two classes are as follows: Her null hypothesis and alternative hypothesis are: (a) Determine the test statistic. Show all work; writing the correct test statistic, without supporting work, will receive no credit. The test statistic under H0 is : F= s12/ s22 = 1.777777778 (b) Determine the P-value for this test. Show all work; writing the correct P-value, without supporting work, will receive no credit. P-value: P(F < -1.777777778) + P(F > 1.777777778) =0.0792 (c) Is there sufficient evidence to justify the rejection of H0 at the significance level of 0.05? Explain. No, there is no sufficient evidence to justify the rejection of H0 since the p-value is greater than the level of significance