1) What is the probability that the random variable has a value less than 3? 1)

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1 Ch 6 and 7 Worksheet Disclaimer; The actual exam differs NOTE: ON THIS TEST YOU WILL NEED TO USE TABLES (NOT YOUR CALCULATOR) TO FIND PROBABILITIES UNDER THE NORMAL OR CHI SQUARED OR T DISTRIBUTION! SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Using the following uniform density curve, answer the question. 1) What is the probability that the random variable has a value less than 3? 1) If Z is a standard normal variable, find the probability. 2) The probability that Z lies between and ) 3) P(Z > 0.59) 3) The Precision Scientific Instrument Company manufactures thermometers that are supposed to give readings of 0 C at the freezing point of water. Tests on a large sample of these thermometers reveal that at the freezing point of water, some give readings below 0 C (denoted by negative numbers) and some give readings above 0 C (denoted by positive numbers). Assume that the mean reading is 0 C and the standard deviation of the readings is 1.00 C. Also assume that the frequency distribution of errors closely resembles the normal distribution. A thermometer is randomly selected and tested. Find the temperature reading corresponding to the given information. 4) Find P 40, the 40th percentile. 4) Assume that X has a normal distribution, and find the indicated probability. 5) The mean is! = 15.2 and the standard deviation is σ = 0.9. Find the probability that X is greater than ) 6) A bank's loan officer rates applicants for credit. The ratings are normally distributed with a mean of 200 and a standard deviation of 50. Find P60, the score which separates the lower 60% from the top 40%. 6) Find the indicated probability. 7) The incomes of trainees at a local mill are normally distributed with a mean of $1100 and a standard deviation of $150. What percentage of trainees earn less than $900 a month? 7) 8) The Los Angeles Beanstalk Club has a height requirement that women must be at least 68 in. tall. Women s heights are normally distributed with a mean of 63.6 in. and a standard deviabon of 2.5 in. What percentage of women meet that requirement? Draw a density curve with all relevant informa/on. 8) 1

2 9) The amount of snowfall falling in a certain mountain range is normally distributed with a mean of 71 inches, and a standard deviation of 10 inches. What is the probability that the mean annual snowfall during 25 randomly picked years will exceed 73.8 inches? 9) 10) A study of the amount of time it takes a mechanic to rebuild the transmission for a 1992 Chevrolet Cavalier shows that the mean is 8.4 hours and the standard deviation is 1.8 hours. If 40 mechanics are randomly selected, find the probability that their mean rebuild time exceeds 8.1 hours. 10) Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution. Hint: for these few questions don't forget to use the continuity correction. 11) With n = 18 and p = 0.30, estimate P(6). 11) 12) Find the critical value zα/2 that corresponds to a degree of confidence of 98%. 12) Express the confidence interval in the form of p^ ± E. 13) < p < ) Find the minimum sample size you should use to assure that your estimate of p^ will be within the required margin of error around the population p. 14) Margin of error: 0.012; confidence level: 93%; p^ and q^ unknown 14) 15) Margin of error: 0.03; confidence level: 99%; from a prior study, p^ is estimated by ) Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p. 16) A survey of 865 voters in one state reveals that 408 favor approval of an issue before the 16) legislature. Construct the 95% confidence interval for the true proportion of all voters in the state who favor approval. Use the given degree of confidence and sample data to construct a confidence interval for the population mean!. Assume that the population has a normal distribution. 17) n = 10, x = 13.2, s = 4.1, 95 percent 17) 18) Thirty randomly selected students took the calculus final. If the sample mean was 85 and the sample standard deviation was 9.2, construct a 99 percent confidence interval for the mean score of all students. 18) 19) Find the critical value χ 2 R 95 percent. corresponding to a sample size of 3 and a confidence level of 19) 2

3 Use the given degree of confidence and sample data to find a confidence interval for the population standard deviation σ. Assume that the population has a normal distribution. 20) Weights of eggs: 95% confidence; n = 22, x = 1.56 oz, s = 0.45 oz 20) Using the following uniform density curve, answer the question. 21) What is the probability that the random variable has a value less than 2.5? 21) 22) What is the probability that the random variable has a value between 0.2 and 0.8? 22) Assume that the weight loss for the first month of a diet program varies between 6 pounds and 12 pounds, and is spread evenly over the range of possibilities, so that there is a uniform distribution. Find the probability of the given range of pounds lost. 23) Between 8 pounds and 11 pounds 23) Find the area of the shaded region. The graph depicts the standard normal distribution with mean 0 and standard deviation 1. 24) 24) z 25) 25) z Find the indicated z score. The graph depicts the standard normal distribution with mean 0 and standard deviation 1. 26) Shaded area is ) z 3

4 27) Shaded area is ) z If z is a standard normal variable, find the probability. 28) The probability that z is greater than ) The Precision Scientific Instrument Company manufactures thermometers that are supposed to give readings of 0 C at the freezing point of water. Tests on a large sample of these thermometers reveal that at the freezing point of water, some give readings below 0 C (denoted by negative numbers) and some give readings above 0 C (denoted by positive numbers). Assume that the mean reading is 0 C and the standard deviation of the readings is 1.00 C. Also assume that the frequency distribution of errors closely resembles the normal distribution. A thermometer is randomly selected and tested. Find the temperature reading corresponding to the given information. 29) Find Q 3, the third quartile. 29) Provide an appropriate response. 30) Assume that adults have IQ scores that are normally distributed with a mean of 100 and a standard deviation of 15 (as on the Wechsler test). Find the probability that a randomly selected adult has an IQ between 90 and 120 (somewhere in the range of normal to bright normal). 30) 31) Find P15, which is the IQ score separating the bottom 15% from the top 85%. 31) Assume that X has a normal distribution, and find the indicated probability. 32) The mean is! = 22.0 and the standard deviation is σ = 2.4. Find the probability that X is between 19.7 and ) 33) The amount of snowfall falling in a certain mountain range is normally distributed with a mean of 94 inches, and a standard deviation of 14 inches. What is the probability that the mean annual snowfall during 49 randomly picked years will exceed 96.8 inches? 33) Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution. Hint: for these few questions don't forget to use the continuity correction. 34) With n = 20 and p = 0.60, estimate P(fewer than 8). 34) 35) Estimate the probability of getting exactly 43 boys in 90 births. 35) Express the confidence interval using the indicated format. 36) Express the confidence interval (0.668, 0.822) in the form of p^ ± E. 36) 4

5 37) The following confidence interval is obtained for a population proportion, p: < p < Use these confidence interval limits to find the margin of error, E. 37) Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p. 38) n = 107, x = 50; 88% confidence 38) 39) n = 110, x = 55; 88% confidence 39) 40) n = 164, x = 122; 95% confidence 40) Use the given data to find the minimum sample size required to estimate the population proportion. 41) Margin of error: 0.04; confidence level: 94%; p^ and q^ unknown 41) 42) Margin of error: 0.04; confidence level: 95%; from a prior study, p^ is estimated by the decimal equivalent of 89%. 42) Use the given degree of confidence and sample data to construct a confidence interval for the population mean!. Assume that the population has a normal distribution. 43) n = 12, x = 28.3, s = 4.8, 99% confidence 43) 44) n = 30, x = 83.1, s = 6.4, 90% confidence 44) 45) A laboratory tested twelve chicken eggs and found that the mean amount of cholesterol was 225 milligrams with s = 15.7 milligrams. Construct a 95% confidence interval for the true mean cholesterol content of all such eggs. 45) 46) A savings and loan association needs information concerning the checking account balances of its local customers. A random sample of 14 accounts was checked and yielded a mean balance of $ and a standard deviation of $ Find a 98% confidence interval for the true mean checking account balance for local customers. 46) 47) Find the critical value χ 2 R 95 percent. corresponding to a sample size of 3 and a confidence level of 47) 48) Find the critical value χ 2 R 98 percent. corresponding to a sample size of 5 and a confidence level of 48) 49) Find the critical value χ 2 L of 95 percent. corresponding to a sample size of 24 and a confidence level 49) 5

6 50) Find the chi-square value χ 2 L corresponding to a sample size of 17 and a confidence 50) level of 98 percent. Use the given degree of confidence and sample data to find a confidence interval for the population standard deviation σ. Assume that the population has a normal distribution. Round the confidence interval limits to the same number of decimal places as the sample standard deviation. 51) College students' annual earnings: 98% confidence; n = 9, x = $3122, s = $876 51) 52) College students' annual earnings: 98% confidence; n = 9, x = $3959, s = $886 52) 53) A sociologist develops a test to measure attitudes about public transportation, and 27 randomly selected subjects are given the test. Their mean score is 76.2 and their standard deviation is Construct the 95% confidence interval for the standard deviation, σ, of the scores of all subjects. 53) 6

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8 Answer Key Testname: 227CH6_7WKSHT 48) ) ) ) $553 < σ < $ ) $559 < σ < $ ) 16.9 < σ <