Question

1. I collect a random sample of size n from a population and from the data collected, I compute a 95% confidence interval for the mean of the population. Which of the following would produce a new confidence interval with larger width (larger margin of error) based on these same data? Circle your answer(s)
   1. Use a smaller confidence level.
   2. Use a larger confidence level.
   3. Use the same confidence level but compute the interval n times.

2. Suppose you know the length of a confidence interval of a population mean is 8.4 and the sample mean (x bar) is 10. Find the margin of error​  , and the​      confidence interval.​

3. We are going to collect a sample data on the average length of stay by renters of our residential properties. We feel comfortable that the population standard deviation is 30 months. We wish to make sure that our margin of error is no more than 5 months. At 95% confidence level, how big is the sample size that we need to collect?  

4. Suppose you have a confidence interval for a population mean but you wish to achieve a narrower interval. Which of the following options would help obtain a narrower the confidence interval? Circle your answer(s)​     
   1. Increase the sample size while keep the confidence level fixed
   2. Decrease the sample size while keep the confidence level fixed
   3. Decrease the confidence level while keep the sample size fixed
   4. Increase the confidence level while keep the sample size fixed

Answer 1

Given:

1) I collect a random sample of size n from a population and from the data collected, I compute a 95% confidence interval for the mean of the population.

Use a larger confidence level

Explaination:

We know that confidence interval is given by

CI = Z/2 × s/√n

Where = sample mean

Z/2 = critical value

s = standard deviation

n = sample size

As we increase the confidence level, the critical value also increases.

This increased critical value results in increasing margin of error given that standard deviation and sample size are constant.

So, width of the confidence interval is increased.

Increased critical value will results in increased margin of error.

So answer is option A

2) Given:

Suppose you know the length of a confidence interval of a population mean is 8.4 and the sample mean (x bar) is 10.

CI = 8.4

= 10

Confidence interval is given by

(+E)-(-E) = CI

(10+E)-(10-E) = 8.4

10+E-10+E = 8.4

2E = 8.4

E = 8.4/2 = 4.2

Margin of error is E = 4.2

Confidence interval is

CI = E

= 10 4.2

= (5.8, 14.2)

Therefore confidence interval is (5.8, 14.2)

3) We are going to collect a sample data on the average length of stay by renters of our residential properties.

Margin of error, E = 5

Standard deviation, = 30

Significance level, = 1-0.95 = 0.05

At 95% confidence level the critical value of Z is

Z/2 = 1.96

The formula for sample size is

n = (Z/2/E) ^2 × 2

= (1.96/5)^2 × 30^2

= 138.29

= 138

So sample size is n = 138

4) Suppose you have a confidence interval for a population mean but you wish to achieve a narrower interval.

Increase the sample size while keep the confidence level fixed.

Answer - option A