Unit 5 Review-Sampling Distributions

A statistic is an unbiased estimator of a parameter when

• in many samples, the values of the statistic are centered at the parameter

• in many samples, the values of the statistic are close to the parameter

• the statistic is calculated from a random sample.

• in a single sample, the value of the statistic is equal to the parameter

The Gallup Poll asked a random sample of 1785 adults whether they attended church during the past week.  Let p-hat be the proportion of people in the sample who attended church.  A newspaper report claims that 40% of all U.S. adults went to church last week.  Suppose this claim is true. 
What is the mean of the sampling distribution of p-hat?

• 0.4
• 0.44
• 0.56
• 0.6

What happens to the shape of a sampling distribution of sample means as n increases?

• It becomes narrower and bimodal.
• It becomes narrower and more normal.
• It becomes wider and skewed right. 
• It becomes wider and more normal. 

Researchers will conduct a study of the television-viewing habits of children. They will select a simple random sample of children and record the number of hours of television the children watch per week. The researchers will report the sample mean as a point estimate for the population mean. Which of the following statements is correct for the sample mean as a point estimator?

• A sample of size 25 will produce more variability of the estimator than a sample of size 50.

• A sample of size 25 will produce less variability of the estimator than a sample of size 50.

• A sample of size 25 will produce a biased estimator, but a sample size of 50 will produce an unbiased estimator.

• A sample of size 25 will produce a more biased estimator than a sample of size 50.

• A sample of size 25 will produce a less biased estimator than a sample of size 50.

The time it takes all cross country runners to complete a course has a mean of 32.1 minutes with a standard deviation of 10.2 minutes. What is the probability that an SRS of 36 runners will have a mean completion time greater than 34 minutes?

• .4261

• .5739

• .8681

• .1319

When x̅ differs from sample to sample, we say this is an example of

• variability among the samples

• bias among the samples

• not having large enough sample sizes

• poor survey techniques

The central limit theorem refers to which of the following characteristics of the sampling distribution of the sample mean?

• Regardless of the shape of the population’s distribution, the sampling distribution of the sample mean from sufficiently large samples will be approximately Normally distributed.

• Regardless of the shape of the population’s distribution, the standard deviation of the sampling distribution of the sample mean from sufficiently large samples will be σ/√n.

• Regardless of the shape of the population’s distribution, the mean of the sampling distribution of the sample mean from sufficiently large samples will be equal to the mean of the population.

• As you take larger and larger samples from a Normally distributed population, the standard deviation of the sampling distribution of the sample mean gets smaller and smaller.

• As you take larger and larger samples from a Normally distributed population, the mean of the sampling distribution of the sample mean gets closer and closer to the population mean.

A student investigating study habits asks a simple random sample of 16 students at her school how many minutes they spent on their English homework the previous night. Suppose the actual parameter values for this variable are μ = 45 minutes and σ = 15 minutes. Which of the following best describes what we know about the sampling distribution of means for the student's sample?

• μx = 45; σx unknown; shape of distribution unknown

• μx = 45; σx = 15; distribution approximately Normal

• μx = 45; σx = 15; shape of distribution unknown

• μx = 45; σx = 3.75; distribution approximately Normal

• μx = 45; σx = 3.75; shape of distribution unknown

In one region of the country, the mean length of stay in hospitals is 5.5 days with standard deviation 2.6 days. Because many patients stay in the hospital for considerably more days, the distribution of length of stay is strongly skewed to the right. Consider random samples of size 100 taken from the distribution with the mean length of stay, x̄, recorded for each sample. Which of the following is the best description of the sampling distribution of x̄?

• Strongly skewed to the right with mean 5.5 days and standard deviation 2.6 days

• Strongly skewed to the right with mean 5.5 days and standard deviation 0.26 day

• Strongly skewed to the right with mean 5.5 days and standard deviation 0.026 day

• Approximately normal with mean 5.5 days and standard deviation 2.6 days

• Approximately normal with mean 5.5 days and standard deviation 0.26 day

In a national study on transportation patterns, 1,000 randomly selected adults will be asked the question: How many trips per week do you make to the grocery store? The sample mean will be computed. Let μ denote the population mean response to the question if everyone in the population is to be asked the question. Is the sample mean X̄ unbiased for estimating μ?

• Yes, because for random samples the mean (expected value) of the sample mean X̄ is equal to the population mean μ.

• Yes, because with a sample size of 1,000 the standard deviation of the sample mean X̄ is small.

• Yes, because the wording of the question is not biased.

• No, because the sample mean X̄ does not always equal the population mean μ.

• No, because number of trips to the grocery store is not normally distributed so the mean (expected value) of the sample mean X̄ does not equal the population mean μ.