Confidence Intervals for Means

Two independent random samples were collected from the same population to estimate the population mean. Sample A had a sample size of 25 and a sample mean of 50. The 95 percent confidence interval constructed from sample A had a margin of error of 4.2. Sample B had a sample size of n
n and a sample mean of x. The k percent confidence interval constructed from sample B had a margin of error of 3.7. Assume both samples had the same sample standard deviation. Which of the following values from sample B explains why the margin of error for sample A is greater than the margin of error for sample B?

• n = 25, x = 50, k = 95

• n = 25, x = 60, k = 95

• n = 25, x = 40, k = 95

• n = 20, x = 50, k = 99

• n = 30, x = 40, k = 99

When using a one-sample t-procedure to construct a confidence interval for the mean of a finite population, a condition is that the population size be at least 10 times the sample size. The reason for the condition is to ensure that

• the sample size is large enough

• the central limit theorem is applicable for the sample mean

• the sample standard deviation is a good approximation of the population standard deviation

• the degree of dependence among observations is negligible

• the sampling method is not biased

Which of the following correctly compares the t-distribution and z-distribution?

• For small sample sizes, the density curve of the t-distribution is not symmetric, but the density curve of the z-distribution is symmetric.

• For large sample sizes, the density curve of the t-distribution is not symmetric, but the density curve of the z-distribution is symmetric.

• The curves of both distributions are symmetric, but the height of the density curve of the t-distribution is taller than the height of the density curve of the z-distribution.

• The area under the density curve of the t-distribution is greater than the area under the density curve of the z-distribution, especially for small sample sizes.

• The density curve of the t-distribution is more spread out than the density curve of the z-distribution, especially for small sample sizes.

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.

A large company is considering opening a franchise in St. Louis and wants to estimate the mean household income for the area using a simple random sample of households. Based on information from a pilot study, the company assumes that the standard deviation of household incomes is σ = $7,200. Of the following, which is the least number of households that should be surveyed to obtain an estimate that is within $200 of the true mean household income with 95 percent confidence?

• 75

• 1,300

• 5,200

• 5,500

• 7,700

A marine biologist wants to estimate the average weight of a population of dolphins living in a certain region of the ocean. The biologist will collect a random sample of dolphins and use the sample weights to create the estimate.

Which of the following is an appropriate method for the biologist to use for inference to the population?

• A one-sample t-interval for a population mean

• A one-sample t-interval for a sample mean

• A one-sample z-interval for a population proportion

• A matched-pairs t-interval for a mean difference

• A two-sample t-interval for a difference between means

Ecologists wanted to estimate the mean biomass (amount of vegetation) of a certain forested region. The ecologists divided the region into plots measuring 1 square meter each, and they selected a random sample of 9 plots. The mean biomass of the 9 plots was 4.3 kilograms per square meter ( kg/m² ) and the standard deviation was 1.5 kg/m² . Assuming all conditions for inference are met, which of the following is a 95 percent confidence interval for the population mean biomass, in kg/m² ?

• $4.3\pm1.96\frac{\sqrt{1.5}}{3}$

• $4.3\pm1.96\frac{1.5}{3}$

• $4.3\pm2.306\frac{\sqrt{1.5}}{9}$

• $4.3\pm2.306\frac{1.5}{9}$

• $4.3\pm2.306\frac{1.5}{3}$

Animal scientists studied foraging behavior of the scrub lizard, found in central Florida. Foraging is the process of searching for food. To study such behavior, the scientists recorded the number of head movements per minute for a sample of 63 lizards. A 95 percent confidence interval constructed from the sample is given as 2.7±0.62

head movements per minute.

Based on the interval, is a claim of 3 head movements per minute plausible?

• The claim is not plausible because 3 head movements per minute is contained within the interval.

• The claim is not plausible because 3 head movements per minute is not contained within the interval.

• The claim is not plausible because 95 percent of 3 is greater than the sample mean of 2.7.

• The claim is plausible because 3 head movements per minute is not contained within the interval.

• The claim is plausible because 3 head movements per minute is contained within the interval.

Monthly rent was determined for each apartment in a random sample of 100 apartments. The sample mean was $820 and the sample standard deviation was $25. An approximate 95 percent confidence interval for the true mean monthly rent for the population of apartments from which this sample was selected is ($815, $825). Which of the following statements is a correct interpretation of the 95 percent confidence level?

• In this population, about 95 percent of all rental prices are between $815 and $825.

• In this sample, about 95 percent of the 100 rental prices are between $815 and $825.

• In repeated sampling, the method produces intervals that include the population mean approximately 95 percent of the time.

• In repeated sampling, the method produces intervals that include the sample mean approximately 95 percent of the time.

• There is a probability of 0.95 that the true mean is between $815 and $825.

The director of fitness for a large corporation with over 5,000 employees recorded the resting heart rate, in beats per minute (bpm), for 35 employees who were known to wear activity trackers. The boxplot summarizes the results. The director wants to estimate the resting heart rate for all employees with a confidence interval. Have all conditions for inference been met?

• Yes, all conditions have been met.

• No, the distribution of the sample data is not approximately symmetric.

• No, the sample size is greater than 10 percent of the population size.

• No, the distribution of resting heart rate in the population cannot be assumed to be approximately normal.

• No, the sample was not selected at random.