The random sample shown below was selected from a normal distribution. Complete

Question

The random sample shown below was selected from a normal distribution. Complete parts a and b.

10,9,3,8,4,2

a. Construct a 90% confidence interval for the population mean .

(     ,     ) (Round to two decimal places as needed.)

b. Assume that sample mean x and sample standard deviation s remain exactly the same as those you just calculated but that are based on a sample of n=25 observations. Repeat part a. What is the effect of increasing the sample size on the width of the confidence intervals?

The confidence interval is (    ,    ) (Round to two decimal places as needed.)

What is the effect of the sample size on the width of the confidence interval?

A. As the sample size increases, the width increases.

B.As the sample size increases, the width stays the same.

C. As the sample size increases, the width decreases.

Explanation / Answer

a. Construct a 90% confidence interval for the population mean .

Solution:

The confidence interval formula for the population mean is given as below:

Confidence interval = Xbar -/+ t*SD/sqrt(n)

From the given data, we have

Sample mean = Xbar = 6

Sample standard deviation = SD = 3.405877273

Sample size = n = 6

Confidence level = 90%

Degrees of freedom = n – 1 = 6 – 1 = 5

Critical t value = 2.0150

(By using t-table or excel)

Confidence interval = 6 -/+ 2.0150*3.405877273/sqrt(6)

Confidence interval = 6 -/+ 2.8018

Lower limit = 6 – 2.8018 = 3.20

Upper limit = 6 + 2.8018 = 8.80

Confidence interval = (3.20, 8.80)

b. Assume that sample mean x and sample standard deviation s remain exactly the same as those you just calculated but that are based on a sample of n=25 observations. Repeat part a.

Solution:

Here, we have to repeat the above confidence interval by changing the value of sample size as n = 25

Sample size = n = 25

Degrees of freedom = n – 1 = 25 – 1 = 24

Critical t value = 1.7109

Confidence interval = 6 -/+ 1.7109*3.405877273/sqrt(25)

Confidence interval = 6 -/+ 1.1654

Lower limit = 6 – 1.1654 = 4.83

Upper limit = 6 + 1.1654 = 7.17

Confidence interval = (4.83, 7.17)

What is the effect of increasing the sample size on the width of the confidence intervals?

Correct Answer:

C. As the sample sizeincreases, the width decreases.

(According to comparison from above two confidence intervals)

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