What is the sampling distribution of phat the sample

1. According to the National Center for Health Statistics (2004), 22.4% of adults are smokers. A random sample of 300 adults is obtained.

(a) What is the sampling distribution of phat, the sample proportion of adults who smoke? Justify your answer.

(b) In a random sample of 300 adults, what is the probability that at least 50 are smokers?

(c) Would it be unusual if a random sample of 300 adults results in 18% or less being smokers? Explain your answer.

2. A machine at K&A Tube & Manufacturing Company produces a certain copper tubing component in a refrigeration unit. The tubing components produced by the manufacturer have a mean diameter of 0.75 inch with a standard deviation of 0.004 inch. The quality-control inspector takes a random sample of 30 components once each week and calculates the mean diameter of these components. If the mean is either less than 0.748 inch or greater than 0.752 inch, the inspector concludes that the machine needs an adjustment.

(a) What is the sampling distribution of the sample mean diameter for a random sample of 30 such components? Justify your answer.

(b) What is the probability that, based on a random sample of 30 such components, the inspector will conclude that the machine needs an adjustment?

3. In a random sample of 678 adult males 20 to 34 years of age, it was determined that 58 of them have hypertension (high blood pressure). Source: The Centers for Disease Control.

(a) Obtain a point estimate for the proportion of adult males 20 to 34 years of age who have hypertension.

(b) Construct a 95% confidence interval for the proportion of adult males 20 to 34 who have hypertension. Interpret the confidence interval.

You wish to conduct your own study to determine the proportion of adult males 20 to 34 years old who have hypertension.

(c) What sample size would be needed for the estimate to be within 3 percentage points (interval length is 0.06) with 95% confidence if you use the point estimate obtained in part (a)?

(d) What sample size would be needed for the estimate to be within 3 percentage points with 95% confidence if you don't have a prior estimate (use phat=.5)?

4. A random sample of 60 married couples who have been married 7 years was asked the number of children they have. The results of the survey are as follows:

 0 0 0 3 3 3 1 3 2 2 3 1 3 2 4 0 3 3 3 1 0 2 3 3 1 4 2 3 1 3 3 5 0 2 3 0 4 4 2 2 3 2 2 2 2 3 4 3 2 2 1 4 3 2 4 2 1 2 3 2

(a) Obtain a point estimate for the mean and standard deviation number of children of all couples who have been married 7 years.

(b) What is the shape of the distribution of the sample mean? Why?

(c) Compute a 95% confidence interval for the mean number of children of all couples who have been married 7 years. Interpret this interval.

(d) Compute a 99% confidence interval for the mean number of children of all couples who have been married 7 years. Interpret this interval.

5. An environmentalist is interested in determining if the pH of the creek water behind his house is affected by the new development upstream. He knows that a neutral stream has a pH of 7. He takes 32 samples of water and determines the pH. Is the pH of the creek significantly different from neutral? Conduct a significance test to test this claim at the 0.05 significant level.

 7.5 7.6 7.1 6.2 6.3 6.9 7.1 7.3 6.3 6.6 7.1 7.1 6.3 6.9 6.7 6.9 7.3 7.5 7 6.1 6.2 6.8 7 7.2 6.4 6.7 7.2 7.2 6.5 7 6.8 6.9

(a) Write the null and alternative hypotheses.

(b) State the null hypothesis, in words.

(c) What is the critical region? (Reject the null if...)

(d) State your conclusion, and interpret.

(e) Determine a 95% CI for Fe and interpret. Compare the results of the confidence interval to those of the significance test at a = 0.05.