Chapter Quiz 399 Important Formulas Confidence interval for a mean, standard deviation Sample size to construct an interval for p with margin of known: error m: ̄x − z��∕2 �� √ n < �� < ̄x + z��∕2 �� √ n n = ̂p(1 − ̂p) (z��∕2 m )2 if a value for ̂p is available Sample size to construct an interval for �� with margin of error m: n = 0.25 (z��∕2 m )2 if no value for ̂p is available n = (z��∕2 ⋅ �� m )2 Confidence interval for the variance of a normal distribution: Confidence interval for a mean, standard deviation unknown: (n − 1)s2 ��2 ��∕2 < ��2 < (n − 1)s2 ��2 1−��∕2 ̄x − t��∕2 s√ n < �� < ̄x + t��∕2 s√ n Confidence interval for the standard deviation of a normal distribution: √ (n − 1)s2 ��2 ��∕2 < �� < (n − 1)s2 ��2 1−��∕2 √ ��Confidence interval for a proportion: ̂p − z��∕2 √ ̂p(1 − ̂p) n < p < ̂p + z��∕2 √ ̂p(1 − ̂p) n Chapter Quiz 1. Define the following terms: a. Point estimate b. Confidence interval c. Confidence level 2. Find the critical value t��∕2 needed to construct a 90% confidence interval for a population mean with sample size 27. 1.706 3. An owner of a fleet of taxis wants to estimate the mean gas mileage, in miles per gallon, of the cars in the fleet. A random sample of 40 cars is followed for one month, and the sample mean gas mileage is 23.2 with a standard deviation of 5.8. Construct a 90% confidence interval for the mean gas mileage in the fleet. (21.7, 24.7) 4. Construct a 95% confidence interval for the population standard deviation �� if a sample of size 20 has standard deviation s = 10. (7.60, 14.61) 5. A cookie manufacturer wants to estimate the length of time that her boxes of cookies spend in the store before they are bought. She visits a sample of 15 supermarkets and determines the number of days since manufacture of the oldest box of cookies in the store. The mean is 54.8 days with a standard deviation of 11.3 days. A dotplot of the data indicates that the assumptions for constructing a confidence interval for the mean are satisfied. Construct a 99% confidence interval for the mean number of days. (46.1, 63.5) 6. A person selects a random sample of 15 credit cards and determines the annual interest rate, in percent, of each. The sample mean is 12.42 with a sample standard deviation of 1.3. Construct a 95% confidence interval for the mean credit card annual interest rate, assuming that the rates are approximately normally distributed. (11.70, 13.14) 7. Construct a 90% confidence interval for the population standard deviation �� if a sample of size 6 has standard deviation s = 22. (14.79, 45.97) Tech: (14.79, 45.96) 8. Find the critical value z��∕2 needed to construct a confidence interval for a population proportion with confidence level 92%. 1.75 Tech: 1.751 9. Find the critical values for a 98% confidence interval using the chi-square distribution with 18 degrees of freedom. 7.015, 34.805 10. The amount of time that a certain cell phone will keep a charge is known to be normally distributed with standard deviation �� = 16 hours. A sample of 40 cell phones had a mean time of 141 hours. Let �� represent the population mean time that a cell phone will keep a charge. a. What is the point estimate of ��? 141 b. What is the standard error of the point estimate? 2.53 11. Refer to Exercise 10. Suppose that a 95% confidence interval is to be constructed for the mean time. a. What is the critical value? 1.96 b. What is the margin of error? 4.958 c. Construct the 95% confidence interval. (136, 146) 12. Refer to Exercise 10. What sample size is necessary so that a 95% confidence interval will have a margin of error of 1 hour? 984 13. In a survey of 802 U.S. adult drivers, 265 state that traffic is getting worse in their community. Construct a 99% confidence interval for the proportion of adult drivers who think that traffic is getting worse. (0.288, 0.373)
navidi_monk_elementary_statistics_2e_ch7-9
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