Exam 8: Sampling Distributions and Estimation

To estimate the average annual expenses of students on books and class materials, a sample of size 36 is taken. The sample mean is $850 and the sample standard deviation is $54. A 99 percent confidence interval for the population mean is:

Which statement is incorrect? Explain.

The distribution of the sample proportion p = x/n is normal when n ≥ 30.

An estimator must be unbiased if you are to use it for statistical analysis.

What is the approximate width of a 90 percent confidence interval for the true population proportion if there are 12 successes in a sample of 25?

The finite population correction factor (FPCF) can be ignored if n = 7 and N = 700.

All estimators are biased since sampling error always exists to some extent.

Using the conventional polling definition, find the margin of error for a customer satisfaction survey of 225 customers who have recently dined at Applebee's.

To estimate a proportion with a 4 percent margin of error and a 95 percent confidence level, the required sample size is over 800.

The Central Limit Theorem says that, if n exceeds 30, the population will be normal.

As n increases, the standard error decreases.

On the basis of a survey of 545 television viewers, a statistician has constructed a confidence interval and estimated that the proportion of people who watched the season premiere of Glee is between .16 and .24. What level of confidence did the statistician use in constructing this interval? Explain carefully, showing all steps in your reasoning.

A 95 percent confidence interval constructed around p will be wider than a 90 percent confidence interval.

Fulsome University has 16,059 students. In a sample of 200 students, 12 were born outside the United States. Construct a 95 percent confidence interval for the true population proportion. How large a sample is needed to estimate the true proportion of Fulsome students who were born outside the United States with an error of ± 2.5 percent and 95 percent confidence? Show your work and explain fully.

In constructing a confidence interval for a mean with unknown variance with a sample of 25 items, Bob used z instead of t. "Well, at least my interval will be wider than necessary, so it was a conservative error," said he. Is Bob's statement correct?

In a large lecture class, the professor announced that the scores on a recent exam were normally distributed with a range from 51 to 87. Using the Empirical Rule μ ± 3σ to estimate σ, how many students would you need to sample to estimate the true mean score for the class with 90 percent confidence and an error of ±2?

For a sample of size 18, the critical values of chi-square for a 99 percent confidence interval for the population variance are:

A consistent estimator for the mean:

The sample standard deviation s is halfway between the lower and upper confidence limits for the population σ (i.e., the confidence interval is symmetric around s).

A sample of size 5 shows a mean of 45.2 and a sample standard deviation of 6.4. The standard error of the sample mean is approximately 2.86.