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Unformatted text preview: Stats 350 Winter 2010 Exam 1 Solutions 1. How old is your car? A recent study was conducted to compare the age of vehicles in a student parking lot versus those in a faculty parking lot at a major university. A random sample of 15 cars was taken from each lot and the age of the car was recorded by taking the current year and subtracting the model year from it. [9 points] a. This is an example of: (circle one) an observational study a designed experiment b. Whether the car’s owner is a faculty member or student plays the role of the: (circle one) explanatory variable response variable Categorical . whose type is: (circle one) Quantitative Discrete Quantitative Continuous A boxplot of the results was produced and is provided below. When appropriate include units in your answers. c. Find the IQR for the ages of student’s cars that were sampled: IQR= 8.5 – 5.5 = 3 years d. The oldest car from those selected from the faculty lot was how old? Final answer: 7 years old e. One member of the faculty at the university saw the study and was disappointed because 75% of the faculty cars that were sampled were newer than his. How many years old is his car? Final answer: 3 years old f. For the student cars that were sampled, the ages varied by about 2.8 years from their mean age, on average. Joe’s car is pretty new, only 3 years old, he was told that his calculated z‐score was ‐1.3. What was the mean car age for the student cars that were sampled? Show your work. ‐1.3 = (3 – mean)/2.8 (3 – mean) = (‐1.3)(2.8) = ‐3.64 mean = 3 – (‐3.64) = 6.64 Final answer: mean = 6.64 years old 1 | P a g e Stats 350 Winter 2010 Exam 1 Solutions 2. Distracted Drivers – The National Highway Traffic Safety Administration reported that over the past year a total of 5,870 fatalities occurred in 5,463 automobile accidents resulting from ‘distracted driving.’ Distracted driving occurs when someone is text messaging at the same time as operating a vehicle. The age of the driver responsible for these crashes was also recorded and the graph at the right shows a summary of these results. [4 points] a. Since the variable driver’s age (in years) is a: (circle one) Categorical Variable Quantitative Variable the graph at the right is a: (circle one) Histogram Bar Graph b. Find the proportion of drivers that were responsible for a distracted driving fatality who were in the 20‐29 age range. Final Answer: 1449/5463 = 0.2652 c. Circle your answer to correctly complete the sentence: larger than smaller than equal to Based on this graph, the mean age is expected to be ______________ the median age. 3. Laptop Ownership – A random sample of college students was selected and 95% confidence interval for the proportion of all college students that own a laptop computer is given by (0.78, 0.88). [5 points] a. What proportion of the sampled college students stated they own a laptop computer? _midpt = 0.83_ b. Which of the following interpretations regarding the confidence interval and 95% level used are correct? Circle all that are correct. • We can be 95% confident that between 78% and 88% of all sampled students own a laptop. • We can be 95% confident that between 78% and 88% of all college students own a laptop. • If many random samples of the same size were selected, then 95% of the sample proportions would be between 0.78 and 0.88. • If many random samples of the same size were selected, then 95% of the time the population proportion of all college students that own a laptop would be between 0.78 and 0.88. • If many random samples of the same size were selected, we would expect 95% of the samples to produce an interval that does contain the population proportion of all college students that own a laptop. c. Suppose a 99% confidence interval were to be constructed based on the same data. Which one of the following could be the 99% margin of error? Circle your answer: 0.04 0.05 0.06 2 | P a g e Stats 350 Winter 2010 Exam 1 Solutions 4. Food Choices – The owner of a multi‐center child‐care facility would like to test the hypothesis that a majority of parents are satisfied with the healthy food choices for meals and snacks at their children’s center, that is, test H0: p = 0.5 versus Ha: p > 0.5, where p is the population proportion of all parents who are satisfied with the healthy food choices. In a random sample of 15 parents at one center, 14 stated they were satisfied. [7 points] a. Using a 5% significance level, conduct the test. Show all work. n is small here, so must do binomial test p‐value = P(X ≥ 14) = P(X = 14) + P(X = 15)] = ⎜ ⎜ ⎛15 ⎞ ⎛15 ⎞ ⎟(0.5)14 (1 − 0.5)1 + ⎜ ⎟(0.5)15 (1 − 0.5) 0 ⎟ ⎜15 ⎟ ⎝14 ⎠ ⎝⎠ = 15(0.000061)(0.5) + 1(0.0000305)(1) = 0.00046 + 0.0000305 = 0.00049 Test statistic: ____X = 14______ p‐value: ___0.00049____________ Yes No Are the results statistically significant at the 5% level? (circle one) b. By choosing a 5% level, the owner accepts a 5% risk of incorrectly concluding a majority of parents are satisfied. True False c. If the study were to be conducted at another child‐care center, which of the following would lead to a higher probability of correctly concluding a majority of parents are satisfied? Circle all that apply. Use n = 10 Use n = 50 Use a = 0.01 Use a = 0.10 5. Environmental Proposal – Consider the following table that summarizes the attitude (in favor, indifferent, or opposed) to a particular environmental proposal and the political party for the 100 U.S. senators. [5 points] In Favor Indifferent Opposed Total Democrat 27 15 18 60 Republican 13 10 17 40 Total 40 25 35 100 a. What is the probability that a randomly selected Democrat will be in favor of the proposal? P(In Favor | Democrat) = 27/60 = 0.45 Final answer: ___ 0.45_____ b. To determine whether being a Democrat was independent from being in favor of the proposal, you would compare the answer to part (a) with another probability. What is the value of that other probability? P(In Favor) = 40/100 = 0.40 Final answer: ___0.40_____ Hence, being a Democrat and being in favor of the proposal are: (circle one) dependent independent. 3 | P a g e Stats 350 Winter 2010 Exam 1 Solutions 6. Overbooking – Beta Airlines states 90 percent of people who purchase the tickets for flight #121 will show up for the actual flight, and the remaining 10 percent will not. So Beta Airlines decides it will sell a total of 225 tickets for a future flight #121, even though the plane could hold up to 205 passengers. Assume that each person who purchases a ticket for this future flight will travel individually, so whether or not an individual shows up for the flight is independent from whether other people show up. Let the variable X represent the number of people who will show up for the future flight among the 225 people who buy the tickets for that flight. [6 points] a. What is the expected number of people who will show up for the future flight? E(X) = np = 225(0.90) = 202.5 Final answer: ____202.5____ b. What is the standard deviation for the number of people who will show up for the future flight? Standard deviation(X) = . . √ . = 4.5 Final answer: ____4.5______ c. What is the approximate probability that there will be a seat available for every passenger who shows up for the future flight? Show your work. ? . . . So the area to the left of Z = 0.56 is 0.7123 Final answer: ____0.7123____ 7. Accepted to Top Choice – In the population of all high‐school students in the U.S., 20% are accepted to the college that they most wanted to go. Suppose we select a random sample of four high‐school students: their names are Bob, Jane, Kelly and Matt. [6 points] a. What is the probability that Bob and Jane are accepted to the colleges they most wanted to go and Kelly and Matt are not accepted to the ones they wanted? P(Bob accepted and Jane accepted and Kelly not accepted and Matt not accepted) = (0.2)(0.2)(0.8)(0.8) Final answer: ____ 0.0256_____ b. Suppose exactly two students among the four students are accepted to the colleges they most wanted. Those two students could be Bob and Jane as in (a), but they could be other pairs of students. How many possible pairs are there? ! ! ! Final answer: _____6_______ c. Let X = the number of students that are accepted to the colleges they most wanted to go among the four selected students. What is the distribution of X? Include the details about the distribution. Final answer: _____Binomial(n = 4, p = 0.2) distribution__________________________ 4 | P a g e Stats 350 Winter 2010 Exam 1 Solutions 8. Determine if each of the following statements is true or false. Clearly circle your answer. [5 points] a. Outliers should always be removed from the data set because they are undesirable and provide no additional information to the analysis. b. Response bias occurs when participants respond differently from how they truly feel. c. When data has been collected over time, a time plot is used to check the identically distributed assumption required for most inference procedures. d. It is possible to get a negative standard deviation, this just means most of the observations fall below the mean value. e. For any given sample size, the sampling distribution of the sample proportion is normal. True True True True True False False False False False 9. Completing that Homework – Suppose that the time it takes for students to complete Stats 350 homework is normally distributed with mean of 60 minutes and standard deviation of 15 minutes. [6 points] a. Write an interpretation of the standard deviation of 15 minutes in terms of an average distance. The completion times for Stats 350 homework vary by about 15 minutes from the mean completion time of 60 minutes, on average. OR The average distance between the completion times for Stats 350 homework and the mean of 60 minutes is approximately 15 minutes. OR On average, the completion times vary from the mean of 60 minutes by roughly 15 minutes. b. Suppose a student is randomly selected from Stats 350 class. What is the probability that it will take more than 70 minutes for the student to complete the homework? Show your work. Final answer: ___0.2514_____ c. Jack, a student in Stats 350, claims that he is one of the top 10% who can finish the homework the fastest. If his claim is true, it would take at most _________________ minutes for him to complete the homework. Fill in the blank. Show your work. Final answer: ____40.8_____
5 | P a g e Stats 350 Winter 2010 Exam 1 Solutions 10. Satisfied with your first‐year experience? – Undergraduate students today are less satisfied with their first year college experience than 5 years ago, a survey says. The survey is based on a representative sample of 500 undergraduates completing their first year of college last May. Fifty‐eight percent of those surveyed reported that they were happy with their first year college experience, down from 63 percent (the established rate 5 years prior). Is the result from the recent survey sufficient evidence to conclude that the ‘satisfied with their first year college experience’ rate is significantly lower than the prior rate of 63%? Use a 5% significance level. [11 points] a. The following definition of the parameter of interest is not complete, so you are asked to correct it. Let p = the population proportion of all first year college students … who were satisfied with their first year college experience. b. State the appropriate hypotheses to be tested. H0: ___ p = 0.63______________ Ha:____ p < 0.63__________ c. It is stated that the sample is representative. Check the remaining condition required for conducting the test of hypotheses in part (a). Show your work. np0 = 500(0.63) = 315 is at least 10 and n(1 – p0) = 500(1 – 0.63) = 185 is at least 10 d. Compute the corresponding test statistic. Show all work. Test Statistic: _______Z = ‐2.32_______________ e. Find the corresponding p‐value. Show your work. f. What would be the statistical decision? (circle one) p‐value: ______0.0102_________________ Fail to reject H0 Reject H0 g. What would be the real‐world conclusion? There (circle one) is is not sufficient evidence to conclude that the ‘satisfied with their first year college experience’ rate is significantly lower than the prior rate of 63%. h. In making your decision in part (f) you could have made a mistake. What is the statistical name for the mistake you could have made? Type I error_____ Name of mistake: 6 | P a g e Stats 350 Winter 2010 Exam 1 Solutions 11. Class Rank – The registrar’s office reports that among all students 34% are Freshmen and 22% are Seniors. [3 points] a. The events F= being a freshman and S = being a senior are: (circle all that apply) probabilities disjoint independent dependent b. Which of the following terms pertain to the values of 34% and 22%? (circle all that apply) parameters statistics p sample proportion population proportion 12. Speaking Spanish – The New Mexico Department of Education is considering expanding the state’s English as a Second Language (ESL) programs. The department first decides to estimate the proportion of state households that speak primarily Spanish at home. In a random sample of 150 state households, a total of 40 households spoke primarily Spanish at home. [4 points] a. Give an estimate of the population proportion of all state households that spoke primarily Spanish. Provide the symbol and value. Estimate: __ ___ = ___40/150 = 0.2667______ symbol value b. Compute a 99% conservative confidence interval for the population proportion of all New Mexico households that speak primarily Spanish. 0.2667 ± . √ 0.2667 ± 0.1052 Confidence Interval: ( ___0.1615___, ___0.3719___) 13. Comparing Test Results – David, Ana, and Namira each took the same test. David received a score of 82 points on the test. Namira was told her score was the 62nd percentile. [4 points] a. Who did better? Or can you tell? Briefly explain your answer. Circle one: David did better, Namira did better, You can’t tell who did better, Because … We don’t have enough information about the distribution of scores. We don’t know what score the 62nd percentile corresponds to; we don’t know the total number of points nor the mean or standard deviation, so we cannot determine if 82 is a good score or if it is a poor score (compared to the 62nd percentile). b. Ana was told her z‐score was ‐1.5. Explain in one brief sentence the meaning of a z‐score of ‐1.5 (in the context of this problem). Ana’s score was 1.5 standard deviation below the mean (or average) score. Note: one cannot assume a normal model for scores, so using table A.1 does not work here. 7 | P a g e ...
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This note was uploaded on 12/10/2010 for the course STATS 250 taught by Professor Gunderson during the Fall '10 term at University of Michigan.
- Fall '10