standard deviation

Due by 8:00 pm 4/13/21. Late submissions may be penalized. Please scan ALL pages as ONE pdf file. Please sign the following Academic Honesty Statement: “This exam is completely my own work. While working on this exam, I did not use my notes, handouts, or textbook, I did not consult anyone else, real or virtual, and I did not use the internet in any manner.” Math 142 (Mowry) Spring 2021 Exam 3 Name . Point Value TOTAL = 100 SHOW ALL YOUR WORK IN THE SPACE PROVIDED! TO RECEIVE FULL CREDIT, YOU MUST DOCUMENT YOUR WORK. [1-10] 2 (i.e. show formulas, numbers, calculator programs, etc.) [11, 12] 3 [15] 9 CIRCLE YOUR FINAL ANSWERS. [13, 16] 8 [14, 17, 18] 5 USE THE ROUND-OFF RULES PRESENTED IN CLASS. . [20, 21] 10 [19, 22] 7 [1-6] TRUE or FALSE? _________[1] Decreasing the sample size will increase the margin of error. _________[2] Increasing the level of confidence will decrease the width of a confidence interval. _________[3] For any given level of confidence, critical t-values are always greater than critical z-values. _________[4] The standard normal distribution has mean = 1 and standard deviation = 0. _________[5] The probability density function for a uniform distribution is a horizontal line. _________[6] If n = 21 and p = 0.35, then the sampling distribution of pˆ will be approximately normal. [7-10] Pick the most correct answer. _________[7] If you want to construct a confidence interval for the mean of a skewed population with unknown standard deviation 𝜎, and your sample size is n = 19, which distribution should be used? (A) Z (B) T (C) none of these _________[8] If you want to construct a confidence interval for the mean of a normally distributed population with unknown standard deviation 𝜎, and your sample size is n = 7, which distribution should be used? (A) Z (B) T (C) none of these _________[9] If you want to construct a confidence interval for the mean of a skewed population with unknown standard deviation 𝜎, and your sample size is n = 41, which distribution should be used? (A) Z (B) T (C) none of these _________[10] If you want to construct a confidence interval for the mean of a normally distributed population with known standard deviation 𝜎, and your sample size is n = 7, which distribution should be used? (A) Z (B) T (C) none of these [11] Find the critical value zα 2 for a 98% confidence interval. (Assume the population data is normally distributed.) € [12] If n = 6, find the critical value tα 2 for a 98% confidence interval. (Assume the population data is normally distributed.) € page 2 [13] Men’s heights are normally distributed with mean 176.5 cm and standard deviation 6.1 cm. The local police department requires that police officers have heights between 170.0 cm and 190.0 cm. (a) Find the percentage of men meeting the height requirement. (b) If the height requirements are changed to exclude the tallest 5% and the shortest 3% of men, what are the new height requirements? [14] An IQ test is designed so that the mean is 100 and the standard deviation is 12 for the population of normal adults. Find the sample size necessary to estimate the mean IQ score of statistics students such that it can be said with 95% confidence that the sample mean is within 2 IQ points of the true mean. page 3 [15] Suppose the time waiting for a bus at the DVC Transit Center is uniformly distributed; wait-time can be anywhere between 0.0 and 13.0 minutes. (a) Find the probability that you must wait 1.0 minute or less. (b) Find the probability that you must wait between 5.0 and 10.0 minutes. (c) Find the probability that you must wait 10.0 minutes or more. Give answers as a decimal rounded-off at the proper place. page 4 [16] Suppose the weights of all DVC students are normally distributed with mean 168.2 pounds and standard deviation 22.1 pounds. (a) If one student is randomly selected, find the probability that the student weighs more than 180 pounds. (b) The elevator in the Math Building can safely lift 1,800 pounds; it is dangerous if the total weight is over 1,800 pounds. If 10 people are in the elevator, find the probability that their mean weight is greater than 180 pounds (hence, the total weight exceeds 1,800 pounds). Is it safe to have 10 people in the elevator? Explain. page 5 [17] Suppose you have collected the following sample data: 942 134 391 158 223 110 Did the sample come from a population that is normally distributed? Explain why or why not and show your work. [18] You plan to conduct a survey to estimate the percentage of adults who have had measles (the disease). Find the number of people who must be surveyed if you want to be 96% confident that the sample percentage is within four percentage points of the true percentage for the population of all adults. [19] In a test of a weight loss program, a sample of 37 adults used a special diet for a month. After the first month, their mean weight loss was 12.3 pounds, with a standard deviation of 5.1 pounds. Construct, and interpret, a 99% confidence interval estimate of the mean weight loss for all such subjects. Express your answer using inequality notation. page 6 [20] The following sample data represent the amount of time (minutes) waiting to be helped in the Math Lab. 2.5 11.6 15.9 0.3 7.3 5.1 (a) Find the mean and standard deviation of the sample data. Round-off your answers appropriately. (b) Did the sample come from a population that is normally distributed? Explain why or why not. (c) Construct a 90% confidence interval estimate for the mean waiting time in the Math Lab. Express your answer using inequality notation. (d) Interpret the meaning of the confidence interval found in part (c). Use complete sentences, proper grammar, and be specific! page 7 [21] In an anonymous survey of 240 students, 42 said that they have cheated on an exam at least once while at DVC. (Of course not you!) (a) Find a point estimate of the proportion of cheaters at DVC. (b) With 95% confidence, calculate the margin of error, E, associated with the point estimate in part (a). (c) Construct a 95% confidence interval estimate for the proportion of all students who have cheated at least once on an exam at DVC. Express your answer using inequality notation. (d) Interpret the meaning of the confidence interval found in part (c). Use complete sentences, proper grammar, and be specific! page 8 [22] Given the population data {7, 11, 18}. Suppose a random sample of n = 2 is selected (with replacement), and the sample mean is calculated. (a) Construct the sampling distribution of the mean. (b) Find the mean and standard deviation of the sampling distribution; use proper symbols for your answers. Do not round-off your answers. (c) Is 𝑥̅ (x-bar, the sample mean) a biased or unbiased estimator of 𝜇? Explain why. EXTRA CREDIT (Total exam score will not exceed 100 points.) Given the population data {7, 11, 18}. Suppose a random sample of n = 2 is selected (with replacement), and the sample proportion of odd numbers is calculated. (a) Construct the sampling distribution of the proportion of odd numbers. (b) Find the mean and standard deviation of the sampling distribution; use proper symbols for your answers. Do not round-off your answers. (c) Is 𝑝̂ (p-hat, the sample proportion) a biased or unbiased estimator of p? Explain why. page 9
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