Sampling Distribution Review

A company has 500 employees and wants to evaluate a proportion using the distribution of sample proportions and normal curve. Under which circumstance will they be able to do this?

• n = 80, p = .2

• n = 40, p = .2

• n = 40, p = .9

• n = 30, p = .4

At a particular college, 78% of all students are receiving some kind of financial aid.The school newspaper selects a random sample of 100 students and 72% of the respondents say they are receiving some sort of financial aid. Which of the following is true?

•  78% is a population and 72% is a sample.
• 72% is a population and 78% is a sample
• 78% is a parameter and 72% is a statistic
• 72% is a parameter and 78% is a statistic

Five estimators for a parameter are being evaluated. The true value of the parameter is 0. Simulations of 100 random samples, each of size n, are drawn from the population. For each simulated sample, the five estimates are computed. The histograms below display the simulated sampling distributions for the five estimators. Which simulated sampling distribution is associated with the best estimator for this parameter?

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Suppose we select an SRS of size n = 100 from a large population having proportion p of successes. Let p̂ be the proportion of successes in the sample. For which value of p would it be safe to use the Normal approximation to the sampling distribution of p̂?

• 0.01

• 0.09

• 0.85

• 0.978

• 0.999

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. 

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, x, recorded for each sample. Which of the following is the best description of the sampling distribution of x ?

• 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

Which of the following pairs of sample size n and population proportion p would produce the greatest standard deviation for the sampling distribution of a sample proportion p?

• n = 1,000 and p close to 0

• n = 1,000 and p close to 1

• n = 1,000 and p close to 1/2

• n = 100 and p close to 0

• n = 100 and p close to 1/2

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.

According to government data, 22 percent of children in the United States under the age of 6 years live in households with incomes that are classified at a particular income level. A simple random sample of 300 children in the United States under the age of 6 years was selected for a study of learning in early childhood. If the government data are correct, which of the following best approximates the probability that at least 27 percent of the children in the sample live in households that are classified at the particular income level? (Note: z represents a standard normal random variable.)

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Employees at a large company can earn monthly bonuses. The distribution of monthly bonuses earned by all employees last year has mean 2.3 and standard deviation 1.3. Let z represent the standard normal distribution. If x represents the mean number of monthly bonuses earned last year for a random sample of 40 employees, which of the following calculations will give the approximate probability that x is less than 2 ?

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