Review problems in sample proportions

Rest of the slides contain problems

Mainly from Chapter 8

Shall do most of them in class

Please bring tables and calculators to class

A simple random sample of 100 athletes is selected from a large high school. In the sample, there are 15 football players. What is the estimate of the standard deviation of the sample proportion of football players?

A simple random sample of 60 blood donors is taken to estimate the proportion of donors with type A blood with a 95% confidence interval. In the sample, there are 10 people with type A blood. What is the margin of error for this confidence interval?

Construct a 97.5% confidence interval. What is the margin of error of this interval

A simple random sample of 85 students is taken from a large university on the West Coast to estimate the proportion of students whose parents bought a car for them when they left for college. When interviewed, 51 students in the sample responded that their parents bought them a car. What is a 95% confidence interval for p, the population proportion of students whose parents bought a car for them when they left for college?

A simple random sample of 100 bags of tortilla chips produced by company X is selected every hour for quality control. In the current sample, 18 bags had more chips (measured in weight) than the labeled quantity. The quality control inspector wishes to use this information to calculate a 90% confidence interval for the true proportion of bags of tortilla chips that contain more than the label states. What is the value of an estimate of standard error of ? What is the 90% confidence interval?

p̂

The Information Technology Department at a large university wishes to estimate p = the proportion of students living in the dormitories who own a computer with a 95% confidence interval. What is the minimum required sample size the IT Department should use to estimate the proportion p with a margin of error no larger than 3 percentage points?

The candy company that makes M&M’s claims that 10% of the M&M’s it produces are green. Suppose that the candies are packaged at random in large bags of 200 M&M’s. When we randomly pick a bag of M&M’s we may assume that this represents a simple random sample of size n = 200. Suppose we wish to test H0: p = 0.10 versus Ha: p ≠ 0.10.

Assuming the null hypothesis what is the sdof the sample proportion of green M&Ms?

(a) Suppose that in the randomly selected bag of M&M’s there are only 12 green M&M’s. What is the value of the large-sample zstatistic?

(b) What is the value of the corresponding P-value?

A noted psychic was tested for extrasensory perception. The psychic was presented with 200 cards face down and asked to determine if the card were one of five symbols: a star, a cross, a circle, a square, or three wavy lines. The psychic was correct in 50 cases. Let p represent the probability that the psychic correctly identifies the symbol on the card in a random trial. Assume the 200 trials can be treated as a simple random sample from the population of all guesses the psychic would make in his lifetime.

Suppose you wished to see if there were evidence that the psychic is doing better than just guessing. To do this, you test the hypotheses H0: p = 0.20 versus Ha: p > 0.20. What is the value of the large-sample zstatistic?

What do we know about the value of the P-value for the hypothesis test?

How large a sample n would you need to estimate p with a margin of error 0.01 with 95% confidence? Use the hypothesized value p = 0.20 as the value for p*.

After once again losing a football game to the college’s archrival, the alumni association conducted a survey to see if alumni were in favor of firing the coach. A simple random sample of 100 alumni from the population of all living alumni was taken. Sixty-four of the alumni in the sample were in favor of firing the coach. Let p represent the proportion of all living alumni who favor firing the coach.

What is a 99% confidence interval for p?

Suppose the alumni association wished to see if the majority of alumni are in favor of firing the coach. To do this they test the hypotheses H0: p = 0.50 versus Ha: p > 0.50. What is the P-value for this hypothesis test?

An inspector inspects large truckloads of potatoes to determine the proportion p in the shipment with major defects prior to using the potatoes to make potato chips. If there is clear evidence that this proportion is less than 0.10, she will accept the shipment. To reach a decision, she will test the hypotheses H0: p = 0.10, Ha: p < 0.10. To do so, she selects a simple random sample. 150 potatoes from a random sample of 3000 potatoes on the truck were defective.

What is the value of the large-sample z statistic?

What is the P-value for this hypothesis test?

Define the hypotheses to test, and the significance level a.

Calculate the value of the test statistic.

Find the P-value based on the observed data.

State the conclusion. ◦ Reject the null hypothesis if the P-value <=a ; if it>a, the

data do not provide sufficient evidence to reject the null.

A recent study of newborns used umbilical cord blood to test for 25 hydroxyvitamin D, which is an indicator of vitamin D status of the baby. It was reported that 65% of babies tested were deficient in vitamin D in spite of the fact that the mothers consumed vitamin D supplements during pregnancy. A researcher in a northern region felt that this percentage was too high for this region because with the reduced hours of sunshine during winter months pregnant women tended to use of higher doses of supplements to compensate. A sample of 125 newborns was tested, and 72 were declared to be deficient in vitamin D.

What would be the appropriate null and alternative hypotheses that the researcher should establish for a test of significance?

Based on the data from this region, what is the value of the appropriate test statistic?

What is the P-value and the conclusion using α = 0.05?

A sociologist is studying the effect of having children within the first two years of marriage on the divorce rate. Using hospital birth records, she selects a simple random sample of 200 couples who had children within the first two years of marriage. Following up on these couples, she finds that 80 couples are divorced within five years. Let p = the population proportion of couples who had children within the first two years of marriage and are divorced within five years.

Suppose the sociologist wishes to test H0: p = ⅓ versus Ha: p ≠ ⅓. At the 10% significance level, are the data statistically significant for testing these hypotheses?