INTRODUCTION TO STATISTICAL THINKING

1.      There are 23 people at a party. Explain what the probability is that any two of them share the same birthday. 

2.      A cold and flu study is looking at how two different medications work on sore throats and fever.  Results are as follows:

·        Sore throat – Medication A:  Success rate – 90% (101 out of 112 trials were successful)

·        Sore throat – Medication B:  Success rate – 83%  (252 out of 305 trials were successful)

·        Fever – Medication A:  Success rate – 71%  (205 out of 288 trials were successful)

·        Fever –  Medication B:  Success rate – 68%  (65 out of 95 trials were successful)

Analyze the data and explain which one would be the better medication for both a sore throat and a fever.

3.      The United States employed a statistician to examine damaged planes returning from bombing missions over Germany in World War II.  He found that the number of returned planes that had damage to the fuselage was far greater than those that had damage to the engines.  His recommendation was to enhance the reinforcement of the engines rather than the fuselages.  If damage to the fuselage was far more common, explain why he made this recommendation.

STAT 200: Introduction to Statistics Final Examination, Fall 2015 OL1/US1

Page 1 of 6 STAT 200 OL1/US1 Sections Final Exam Fall 2015 The final exam will be posted at 12:01 am on October 9, and it is due at 11:59 pm on October 11, 2015. Eastern Time is our reference time. 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. 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. Record your answers and work on the separate answer sheet provided. This exam has 200 total points. 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 Final Examination, Fall 2015 OL1/US1 Page 2 of 6 1. True or False. Justify for full credit. (15 pts) (a) If the variance of a data set is zero, then all the observations in this data set are zero. (b) If P(A) = 0.4 , P(B) = 0.5, and A and B are disjoint, then P(A AND B) = 0.9. (c) Assume X follows a continuous distribution which is symmetric about 0. If , then . (d) A 95% confidence interval is wider than a 90% confidence interval of the same parameter. (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 . 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 Time (in minutes) Frequency Relative Frequency 1.0 – 1.9 3 2.0 – 2.9 12 3.0 – 3.9 0.20 4.0 – 4.9 3 5.0 -5.9 Total 25 2. Complete the frequency table with frequency and relative frequency. Express the relative frequency to two decimal places. (5 pts) 3. What percentage of the checkout times was at least 3 minutes? (3 pts) 4. In what class interval must the median lie? Explain your answer. (5 pts) 5. Does this distribution have positive skew or negative skew? Why? (2 pts) 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) STAT 200: Introduction to Statistics Final Examination, Fall 2015 OL1/US1 Page 3 of 6 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) (5 pts) 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. (4 pts each) 8. Which quiz has less interquartile range in grade distribution? 9. Which quiz has the greater percentage of students with grades 90 and over? 10. Which quiz has a greater percentage of students with grades less than 60? 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. 11. What is the probability that a randomly selected student has neither device? (10 pts) 12. What is the probability that a randomly selected student has a laptop, given that he/she has a tablet? (5 pts) 13. 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? (5 pts) 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) (5 pts) 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. (a) Construct a table describing the probability distribution. (5 pts) (b) Determine the mean and standard deviation of x. (Round the answer to two decimal places) (10 pts) STAT 200: Introduction to Statistics Final Examination, Fall 2015 OL1/US1 Page 4 of 6 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? (5 pts) (b) Find the probability that that she returns at least 1 of the 10 serves from her opponent. (Show work) (10 pts) 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 jalapeño 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 jalapeño fruit is between 1.5 and 4 inches long? (5 pts) 18. Find the 90 th percentile of the jalapeño fruit length distribution. (5 pts) 19. If a random sample of 100 mature jalapeño fruits is selected, what is the standard deviation of the sample mean? (5 pts) 20. A random sample of 100 light bulbs has a mean lifetime of 3000 hours. Assume that the population standard deviation of the lifetime is 500 hours. Construct a 95% confidence interval estimate of the mean lifetime. Show all work. Just the answer, without supporting work, will receive no credit. (8 pts) 21. Consider the hypothesis test given by : 0.5 : 0.5 1 0   H p H p In a random sample of 100 subjects, the sample proportion is found to be p ˆ  0.45 . (a) Determine the test statistic. Show all work; writing the correct test statistic, without supporting work, will receive no credit. (b) Determine the P-value for this test. Show all work; writing the correct P-value, without supporting work, will receive no credit. (c) Is there sufficient evidence to justify the rejection of H0 at the   0.01 level? Explain. (15 pts) STAT 200: Introduction to Statistics Final Examination, Fall 2015 OL1/US1 Page 5 of 6 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? Reaction Time (seconds) Subject Before After 1 6 7 2 8 8 3 4 6 4 7 8 5 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. (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. Show all work; writing the correct P-value, without supporting work, will receive no credit. (d) Is there sufficient evidence to support the claim that consumption of 2 ounces of alcohol increases mean reaction time? Justify your conclusion. (15 pts) 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 1 2 3 4 Number of Bags 150 110 130 110 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. (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 the test. Show all work; writing the correct P-value, without supporting work, will receive no credit. (d) Is there sufficient evidence to support the manager’s claim that the four types are equally popular? Justify your answer. (15 pts) STAT 200: Introduction to Statistics Final Examination, Fall 2015 OL1/US1 Page 6 of 6 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 0 1 3 5 y 1 2 3 8 (a) Find an equation of the least squares regression line. Show all work; writing the correct equation, without supporting work, will receive no credit. (10 pts) (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. (5 pts) 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. (b) Determine the P-value for this test. Show all work; writing the correct P-value, without supporting work, will receive no credit. (c) Is there sufficient evidence to justify the rejection of H0 at the significance level of 0.05? Explain. (10 pts)

Background Information:

In Chapter 4, we examine algorithms.  An algorithm is a step-by-step method of carrying out a procedure to acheive a desired result.  In this activity, we will explore an algorithm that allows us to find the day of the week on which a particular date occured.  As you know, the day of the week on which a date occurs varies from year to year.  One of the factors that affects the placement of days of the week is whether or not a year happens to be a leap year.  In order to use the algorithm, you will need to know about leap years. 

As explained on page 176 of your Mathematical Ideas textbook, “In general, if a year is divisible (evenly) by 4, it is a leap year.  However, there are exceptions.  Century years, such as 1800 and 1900, are not leap years despite the fact that they are divisible by 4.  Furthermore, an exception to the exception, a century year that is divisible by 400 (such as the year 2000) is a leap year.”  Since none of the years above are century years, you can use the divisibility by 4 method to determine if a leap year is involved, or, for a full list of leap years from 1800 to 2400, you can visit the Kalendar-365 Leap Years website. Your answer to item 1 is the number of days in the year you chose.

Your job is to follow the steps in the algorithm to determine the day of the week on which  one of the following important historical events occured. 

The Bombing of Pearl Harbor – December 7, 1941

Assassination of John F. Kennedy – November 22, 1963

Bicentennial of the United States – July 4, 1976

Y2K – January 1, 2000

Terrorist attacks on the United States – September 11, 2001

A Perpetural Calendar Algorithm – developed by Dan Foley, University of New Orleans

The algorithm requires several key numbers.  The key numbers for the month, day, and century are determined by the following tables. 

The following table contains the steps for the algorithm in the left column and an example in the right column.  The example is the date October 10, 1942.

INSTRUCTIONS:  To get full credit for this activity, choose one of the following historical dates, follow the instructions below. 

The Bombing of Pearl Harbor – December 7, 1941

Assassination of John F. Kennedy – November 22, 1963

Bicentennial of the United States – July 4, 1976

Y2K – January 1, 2000

Terrorist attacks on the United States – September 11, 2001

1.  Using the date that you chose,  complete the following table. 

2. Create a post to the Discussion Board with the date you chose as the Subject of your post.

(for example, the subject of your post might be: “Assassination of John F. Kennedy – November 22, 1963″)

3.  Copy and paste your table with the steps you followed and the results column into your post.

4.  Submit your post.  Be sure to subscribe to the thread so that you will be notified if anyone replies to your post.

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