Confidence Intervals - Chap 8 - MC

A random sample of the costs of repair jobs at a large muffler repair shop produces a mean of $127.95 and a standard deviation of $24.03. If the size of this sample is 40, which of the following is an approximate 90 percent confidence interval for the average cost of a repair at this repair shop?

• $127.95 ± $4.87

• $127.95 ± $6.25

• $127.95 ± $7.45

• $127.95 ± $30.81

• $127.95 ± $39.53

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A random sample has been taken from a population. A statistician, using this sample, needs to decide whether to construct a 90 percent confidence interval for the population mean or a 95 percent confidence interval for the population mean. How will these intervals differ?

• The 90 percent confidence interval will not be as wide as the 95 percent confidence interval.

• The 90 percent confidence interval will be wider than the 95 percent confidence interval.

• Which interval is wider will depend on how large the sample is.

• Which interval is wider will depend on whether the sample is unbiased.

• Which interval is wider will depend on whether a z-statistic or a t-statistic is used.

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A simple random sample produces a sample mean, x̄, of 15. A 95 percent confidence interval for the corresponding population mean is 15 ± 3. Which of the following statements must be true?

• Ninety-five percent of the population measurements fall between 12 and 18.

• Ninety-five percent of the sample measurements fall between 12 and 18.

• If 100 samples were taken, 95 of the sample means would fall between 12 and 18.

• P(12 ≤ x̄ ≤ 18) = 0.95

• If μ = 19, this x̄ of 15 would be unlikely to occur.

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A 95 percent confidence interval for the mean time, in minutes, for a volunteer fire company to respond to emergency incidents is determined to be (2.8, 12.3). Which of the following is the best interpretation of the interval?

• Five percent of the time, the time for response is less than 2.8 minutes or greater than 12.3 minutes.

• The probability is 0.95 that a randomly selected time for response will be between 2.8 minutes and 12.3 minutes.

• Ninety-five percent of the time the mean time for response is between 2.8 minutes and 12.3 minutes.

• We are 95% confident that the mean time for response is between 2.8 minutes and 12.3 minutes.

• We are 95% confident that a randomly selected time for response will be between 2.8 minutes and 12.3 minutes.

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A test engineer wants to estimate the mean gas mileage μ (in miles per gallon) for a particular model of automobile. Eleven of these cars are subjected to a road test, and the gas mileage is computed for each car. A dotplot of the 11 gas-mileage values is roughly symmetrical and has no outliers. The mean and standard deviation of these values are 25.5 and 3.01, respectively. Assuming that these 11 automobiles can be considered a simple random sample of cars of this model, which of the following is a correct statement?

• A 95% confidence interval for μ is 25.5 ± 2.228 × 3.01/√11.

• A 95% confidence interval for μ is 25.5 ± 2.201 × 3.01/√11.

• A 95% confidence interval for μ is 25.5 ± 2.228 × 3.01/√10.

• A 95% confidence interval for μ is 25.5 ± 2.201 × 3.01/√10.

• The results cannot be trusted; the sample is too small.

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A school administrator is interested in estimating the proportion of students in the district who participate in community service activities. From a random sample of 100 students in the district, the administrator will construct a 99 percent confidence interval for the proportion of all district students who participate in community service activities. Which of the following statements must be true?

• The population proportion will be in the confidence interval.

• The probability that the confidence interval will include the population proportion is 0.99.

• The probability that the confidence interval will include the sample proportion is 0.99.

• The population proportion and the sample proportion will be equal.

• The probability that the population proportion and the sample proportion will be equal is 0.99.

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An engineer for the Allied Steel Company has the responsibility of estimating the mean carbon content of a particular day's steel output, using a random sample of 15 rods from that day's output. The actual population distribution of carbon content is not known to be normal, but graphic displays of the engineer's sample results indicate that the assumption of normality is not unreasonable. The process is newly developed, and there are no historical data on the variability of the process. In estimating this day's mean carbon content, the primary reason the engineer should use a t-confidence interval rather than a z-confidence interval is because the engineer

• is estimating the population mean using the sample mean

• is using the sample variance as an estimate of the population variance

• is using data, rather than theory, to judge that the carbon content is normal

• is using data from a specific day only

• has a small sample, and a z-confidence interval should never be used with a small sample

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A large city newspaper periodically reports the mean cost of dinner for two people at restaurants in the city. The newspaper staff will collect data from a random sample of restaurants in the city and estimate the mean price using a 90 percent confidence interval. In past years, the standard deviation has always been very close to $35. Assuming that the population standard deviation is $35, which of the following is the minimum sample size needed to obtain a margin of error of no more than $5?

• 90

• 112

• 133

• 147

• 195

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A student working on a history project decided to find a 95 percent confidence interval for the difference in mean age at the time of election to office for former American Presidents versus former British Prime Ministers. The student found the ages at the time of election to office for the members of both groups, which included all of the American Presidents and all of the British Prime Ministers, and used a calculator to find the 95 percent confidence interval based on the t-distribution. This procedure is not appropriate in this context because

• the sample sizes for the two groups are not equal

• the entire population was measured in both cases, so the actual difference in means can be computed and a confidence interval should not be used

• elections to office take place at different intervals in the two countries, so the distribution of ages cannot be the same

• ages at the time of election to office are likely to be skewed rather than bell-shaped, so the assumptions for using this confidence interval formula are not valid

• ages at the time of election to office are likely to have a few large outliers, so the assumptions for using this confidence interval formula are not valid

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A random sample of 432 voters revealed that 100 are in favor of a certain bond issue. A 95 percent confidence interval for the proportion of the population of voters who are in favor of the bond issue is

• 100 ± 1.96 * √(0.5 * 0.5 / 432)

• 100 ± 1.645 * √(0.5 * 0.5 / 432)

• 100 ± 1.96 * √(0.231 * 0.769 / 432)

• 0.231 ± 1.96 * √(0.231 * 0.769 / 432)

• 0.231 ± 1.645 * √(0.231 * 0.769 / 432)

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