Question

All parts please

1) According to a Field Poll, 79% of California adults (actual results are 400 out of 506 surveyed) feel that “education and our schools” is one of the top issues facing California. We wish to construct a 90% confidence interval for the true proportion of California adults who feel that education and the schools is one of the top issues facing California.

a) The margin of error for a 90% confidence level is:_____ (Write as a decimal rounded to the 4th decimal place)

b) A 90% confidence interval for the population proportion is: (_____ ,_____) (Write as a decimal rounded to the 3rd decimal place.)

c) If we changed to an 85% confidence level, what would happen to your confidence interval and your margin of error?

d) If we changed to a 95% confidence level, what would happen to your confidence interval and your margin of error?

e) Five hundred eleven (511) homes in a certain southern California community are randomly surveyed to determine if they meet minimal earthquake preparedness recommendations. One hundred seventy-three (173) of the homes surveyed met the minimum recommendations for earthquake preparedness and 338 did not. Find the confidence interval at the 90% Confidence Level for the true population proportion of southern California community homes meeting at least the minimum recommendations for earthquake preparedness. (_____ , _____) (Round to the 4th decimal place)

f) State the previous confidence interval in the form of p^±ME. _____ ± _____ (Round to the 5th decimal place)

g) Five hundred eleven (511) homes in a certain southern California community are randomly surveyed to determine if they meet minimal earthquake preparedness recommendations. One hundred seventy-three (173) of the homes surveyed met the minimum recommendations for earthquake preparedness and 338 did not. The point estimate for the population proportion of homes that do not meet the minimum recommendations for earthquake preparedness is _____ (Round to the 4th decimal place)

MULTIPLE CHOICE

What is meant by the term “90% confident” when constructing a confidence interval for a proportion?

A.

If we took repeated samples, approximately 90% of the samples would produce the same confidence interval.

B.

If we took repeated samples, approximately 90% of the confidence intervals calculated from those samples would contain the sample proportion.

C.

If we took repeated samples, approximately 90% of the confidence intervals calculated from those samples would contain the true value of the population proportion.

D.

If we took repeated samples, the sample proportion would equal the population mean in approximately 90% of the samples.

Answer 1

1) We are given that:

The sample standard deviation can be approximated to be equal to

The sampling distribution will have the same mean as the population mean (which is unknown). Also, the standard deviation of the sampling distribution is given by:

A 90% confidence interval corresponds to a range of (0.05, 0.95). Using a z-table, we get the corresponding z-score as (-1.65, 1.65). So, we have:

So, the margin of error for a 90% confidence interval is 0.0298.

b) So, the 90% confidence interaval for the population mean will be:

So, a 90% confidence interval for the population proportion is (0.761, 0.820).

c) For 85% confidence interval, the corresponding range is (0.075, 0.925). Using a z-table, we get the z-score as (-1.44, 1.44). So, we have margin of error . So, we have: . So, a 85% confidence interval for the population proportion is (0.764, 0.817). So, the margin of error decreases and the confidence interval decreases when the confidence level is decreased to 85%.

d) For 95% confidence interval, the corresponding range is (0.025, 0.975). Using a z-table, we get the z-score as (-1.96, 1.96). So, we have margin of error . So, we have: . So, a 95% confidence interval for the population proportion is (0.755, 0.826). So, the margin of error increases and the confidence interval increases when the confidence level is increased to 95%.

e) We have:

Sample proportion

Standard deviation

Sample standard deviation

For 90% confidence interval, the range is (0.05, 0.95). So, the z-score is (-1.65, 1.65).

So, the confidence interval will be

f) As written above, we can also write it as:

g) The point estimate is directly calculated from the sample we have gathered. So, the point estimate = 338/511 = 0.6614.

h) What is meant by the term "90% confident" when constructing a confidence interval for a proportion?

Answer: C. If we took repeated samples, approximately 90% of the confidence intervals calculated from those samples would contain the true value of the population proportion.