Question

the random sample shown below was selected from a normal distribution
7, 6 , 5 , 6 , 9 , 3

a.) construct a 90% confidence interval for the population. mean U
(__,__)

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

Answer 1

Part a

Solution:

Confidence interval for Population mean is given as below:

Confidence interval = Xbar ± t*S/sqrt(n)

From given data, we have

Xbar = 6

S = 2

n = 6

df = n – 1 = 5

Confidence level = 90%

Critical t value = 2.0150

(by using t-table)

Confidence interval = Xbar ± t*S/sqrt(n)

Confidence interval = 6 ± 2.0150*2/sqrt(6)

Confidence interval = 6 ± 1.6453

Lower limit = 6 - 1.6453 = 4.35

Upper limit = 6 + 1.6453 = 7.65

Lower limit = 4.35

Upper limit = 7.65

Part b

Solution:

Confidence interval for Population mean is given as below:

Confidence interval = Xbar ± t*S/sqrt(n)

From given data, we have

Xbar = 6

S = 2

n = 25

df = n – 1 = 24

Confidence level = 90%

Critical t value = 1.7109

(by using t-table)

Confidence interval = Xbar ± t*S/sqrt(n)

Confidence interval = 6 ± 1.7109*2/sqrt(25)

Confidence interval = 6 ± 0.6844

Lower limit = 6 - 0.6844 = 5.32

Upper limit = 6 + 0.6844= 6.68

Lower limit = 5.32

Upper limit = 6.68

When we increases the sample size n, then the width of the confidence interval increases.