STT 315 - Lecture 3 Instructor: Aylin ALIN 04/02/2014 Midterm # 2 A Name PID Section # (enrolled) * The exam is closed book and 80 minutes. * You may use a calculator and the formula sheet that you brought to the exam. * Tables for the standard normal distribution and student-t distribution are attached. * The exam has 2 parts including 30 questions in total. * Total points possible for this exam is 0. * Answers recorded on the scatron and not on the test paper are the basis for scoring the exam. Use a pencil to mark your scatron. * Turn in your scatron when you exit the room. You may keep your exam paper and formula sheet. Part I: This part has True-False Questions. Each question is 2 points. 1) A statistic is biased if the mean of the sampling distribution is equal to the parameter it is intended to estimate. 1) 2) The probability of success, p, in a binomial experiment is a parameter, while the mean and standard deviation, μ and σ, are statistics. 2) 3) The sampling distribution of a sample statistic calculated from a sample of n measurements is the probability distribution of the parameter. 3) 4) If x is a good estimator for μ, then we expect the values of x to cluster around μ. 4) 5) The minimum-variance unbiased estimator (MVUE) has the least variance among all unbiased estimators. 5) 6) As the sample size gets larger, the standard error of the sampling distribution of the sample mean gets larger as well. 6) 7) Parking at a large university can be extremely difficult at times. One particular university is trying to determine the location of a new parking garage. As part of their research, officials are interested in estimating the average parking time of students from within the various colleges on campus. The parameter of the interest for this study is the population mean, μ. 7) 8) The Central Limit Theorem is important in statistics because for any population, it says the sampling distribution of the sample mean is approximately normal, regardless of the sample size. 8) 1
9) The sampling distribution for p^ is approximately normal for a large sample size n, where n is considered large if both n p^ 30 and n(1 - p^) 30. 9) ) One way of reducing the width of a confidence interval is to reduce the confidence level. ) Part II: This part has 20 Multiple-Choice Questions. Each question is 4 points. 11) A 90% confidence interval for the average salary of all CEOs in the electronics industry was constructed using the results of a random survey of 45 CEOs. The interval was ($1,389 ; $128,192 Give a practical interpretation of the interval. A) 90% of the sampled CEOs have salaries that fell in the interval $1,389 to $128,192. B) 90% of all CEOs in the electronics industry have salaries that fall between $1,389 to $128,192. C) We are 90% confident that the mean salary of all CEOs in the electronics industry falls in the interval $1,389 to $128,192. D) We are 90% confident that the mean salary of the sampled CEOs falls in the interval $1,389 to $128,192. 11) 12) It is desired to estimate the proportion of college students who feel a sudden relief now that their statistics class is over. How many students must be sampled in order to estimate the true proportion to within 2% at the 90% confidence level? A) 1692 B) 2401 C) Cannot determine because no estimate of p or q exists in this problem D) 133 12) 13) The daily revenue at a university snack bar has been recorded for the past five years. Records indicate that the mean daily revenue is $1200 and the standard deviation is $270. The distribution is skewed to the left due to several high volume days (football game days). Suppose that 81 days are randomly selected and the average daily revenue computed. Which of the following describes the sampling distribution of the sample mean? A) normally distributed with a mean of $1200 and a standard deviation of $30 B) normally distributed with a mean of $1200 and a standard deviation of $270 C) normally distributed with a mean of $133 and a standard deviation of $30 D) skewed to the left with a mean of $1200 and a standard deviation of $270 13) 14) Sales of a new line of athletic footwear are crucial to the success of a company. The company wishes to estimate the average weekly sales of the new footwear to within $300 with 90% reliability. The initial sales indicate that the standard deviation of the weekly sales figures is approximately $10. How many weeks of data must be sampled for the company to get the information it desires? A) 11 weeks B) 7 weeks C) 23 weeks D) 37 weeks 14) 15) A marketing research company is estimating the average total compensation of CEOs in the service industry. Data were randomly collected from 18 CEOs and the 95% confidence interval for the mean was calculated to be ($2,181,260, $5,836,180). What additional assumption is necessary for this confidence interval to be valid? A) None. The Central Limit Theorem applies. B) The population of total compensations of CEOs in the service industry is approximately normally distributed. C) The sample standard deviation is less than the degrees of freedom. D) The distribution of the sample means is approximately normal. 15) 2
16-17 A filling operation for a soft drink company fills cans with an amount x which is normally distributed with mean 12.1 ounces and standard deviation of 0.1 ounces. 16) What is the probability that average fill in a random sample of n = 4 cans will be less than 12 ounces? A) 0.05 B) 0.84 C) 0.023 D) 0.16 16) 17) With this probability model, what is the probability that a fill is less than the label claim of 12 ounces? A) 0.03 B) 0.16 C) 0.34 D) 0.06 17) 18) A random sample of 4000 U.S. citizens yielded 2250 who are in favor of gun control legislation. Estimate the true proportion of all Americans who are in favor of gun control legislation using a 99% confidence interval. A).4375 ±.6337 B).5625 ±.0202 C).4000 ±.0202 D).2250 ±.6337 18) 19) For the following t-interval estimate for population mean based a sample of size 19, what is the 19) confidence level? x ± 1.33 s n A) 95% B) 80% C) 98% D) 90% 20) A study was conducted to determine what proportion of all college students considered themselves as full-time students. A random sample of 300 college students was selected and 2 of the students responded that they considered themselves full-time students. A computer program was used to generate the following 95% confidence interval for the population proportion: (0.64814, 0.75186). The sample size that was used in this problem is considered a large sample. What criteria should be used to determine if n is large? A) If n > 30, then n is considered large. B) Both np^ 15 and nq^ 15. C) When working with proportions, any n is considered large. D) If n > 25, then n is considered large. 20) 21) The sampling distribution of the sample mean is shown below. Find the expected value of the sampling distribution of the sample mean. 21) x 2 4 6 p(x) 0.6 0.2 0.2 A) 4 B) 2.2 C) 3 D) 3.2 22) The probability distribution shown below describes a population of measurements. 22) x 1 3 4 6 8 p(x) 0.1 0.2 0.3 0.2 0.2 Suppose that we took repeated random samples of n = 2 observations from the population described above. What is the probablity that x will be 2; i.e, P(x = 2.00)? A) 0.04 B) 0.02 C) 0.15 D) 0.3 23) A university dean is interested in determining the proportion of students who receive some sort of financial aid. Rather than examine the records for all students, the dean randomly selects 196 students and finds that 8 of them are receiving financial aid. Use a 95% confidence interval to estimate the true proportion of students who receive financial aid. A) 0.05 ± 0.0302 B) 0.041 ± 0.0278 C) 0.041 ± 0.0233 D) 0.05 ± 0.0256 23) 3
24) An educator wanted to look at the study habits of university students. As part of the research, data was collected for three variables - the amount of time (in hours per week) spent studying, the amount of time (in hours per week) spent playing video games and the GPA - for a sample of 20 male university students. As part of the research, a 95% confidence interval for the average GPA of all male university students was calculated to be: (2.95, 3.). The researcher claimed that the average GPA of all male students exceeded 2.94. Using the confidence interval supplied above, how do you respond to this claim? A) We are 0% confident that the researcher is incorrect. B) We are 95% confident that the researcher is correct. C) We cannot make any statement regarding the average GPA of male university students at the 95% confidence level. D) We are 95% confident that the researcher is incorrect. 24) 25) Find the value of t0 such that the following statement is true: P(-t0 t t0) =.90 where df = 14. A) 2.145 B) 2.624 C) 1.761 D) 1.345 25) 26) What is the confidence level of the following confidence interval for μ? x ± 1.15 σ n 26) A) 75% B) 98% C) 67% D) 37.5% 27) A random sample of n = 600 measurements is drawn from a binomial population with probability of success.08. Give the mean and the standard deviation of the sampling distribution of the sample proportion, p^. A).92;.003 B).08;.011 C).92;.011 D).08;.003 28) How much money does the average professional football fan spend on food at a single football game? That question was posed to ten randomly selected football fans. The sampled results show that the sample mean and sample standard deviation were $70.00 and $17.50, respectively. Use this information to create a 95 percent confidence interval for the population mean. A) 70 ± 2.262 17.50 C) 70 ± 1.833 17.50 B) 70 ± 2.228 17.50 D) 70 ± 1.960 17.50 27) 28) 29) The length of time a traffic signal stays green (nicknamed the "green time") at a particular intersection follows a normal probability distribution with a mean of 200 seconds and the standard deviation of seconds. Use this information to answer the following questions. Which of the following describes the derivation of the sampling distribution of the sample mean? A) A single sample of sufficiently large size is randomly selected from the population of "green times" and its probability is determined. B) The mean and median of a large randomly selected sample of "green times" are calculated. Depending on whether or not the population of "green times" is normally distributed, either the mean or the median is chosen as the best measurement of center. C) The means of a large number of samples of size n randomly selected from the population of "green times" are calculated and their probabilities are plotted. D) The standard deviations of a large number of samples of size n randomly selected from the population of "green times" are calculated and their probabilities are plotted. 29) 30) The director of a hospital wishes to estimate the mean number of people who are admitted to the emergency room during a 24-hour period. The director randomly selects 49 different 24-hour periods and determines the number of admissions for each. For this sample, x = 17.2 and s 2 = 25. Estimate the mean number of admissions per 24-hour period with a 90% confidence interval. A) 17.2 ±.643 B) 17.2 ±.168 C) 17.2 ± 1.175 D) 17.2 ± 5.875 30) 4
Answer Key Testname: MIDTERM-2-A 1) B 2) B 3) B 4) A 5) A 6) B 7) A 8) B 9) B ) A 11) C 12) A 13) A 14) D 15) B 16) C 17) B 18) B 19) B 20) B 21) D 22) A 23) A 24) B 25) C 26) A 27) B 28) A 29) C 30) C 5