Question

The following random sample was selected from a normal distribution: 4, 6, 3, 5, 9, 3.

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

b. Construct a 95% confidence interval for the population mean.

c. Construct a 99% confidence interval for the population mean.

d. Assume that the sample mean x and sample standard deviation s remain exactly the same as those you just calculated but are based on a sample of n = 25 observations rather than n = 6 observations. Repeat parts a–c. What is the effect of increasing the sample size on the width of the confidence intervals?

e. Use Rejection region to test the null hypothesis that the mean of the population is 6 against the alternative hypothesis, ?<6. Use ? = .05.

f. Use p-Value to test the null hypothesis that the mean of the population is 6 against the alternative hypothesis, ? ≠6. Use ? = .05.

Answer 1

(a)

From the given data, the following statistics are calculated:

n = 6

= 5

s = 2.2804

df = 6 - 1 = 5

= 0.10

From Table, critical values of t = 2.015

Confidence Interval:

So,

Answer is:

(3.124, 6.876)

(b)

From the given data, the following statistics are calculated:

n = 6

= 5

s = 2.2804

df = 6 - 1 = 5

= 0.05

From Table, critical values of t = 2.571

Confidence Interval:

So,

Answer is:

(2.607, 7.393)

(c)

From the given data, the following statistics are calculated:

n = 6

= 5

s = 2.2804

df = 6 - 1 = 5

= 0.01

From Table, critical values of t = 4.032

Confidence Interval:

So,

Answer is:

(1.246, 8.754)

(d)

(i)

From the given data, the following statistics are calculated:

n = 25

= 5

s = 2.2804

df = 25 - 1 = 24

= 0.10

From Table, critical values of t = 1.711

Confidence Interval:

So,

Answer is:

(4,220, 5.780)

(ii)

From the given data, the following statistics are calculated:

n = 25

= 5

s = 2.2804

df = 25 - 1 = 24

= 0.05

From Table, critical values of t = 2.064

Confidence Interval:

So,

Answer is:

(4.059, 5.941)

(iii)

From the given data, the following statistics are calculated:

n = 25

= 5

s = 2.2804

df = 25 - 1 = 24

= 0.01

From Table, critical values of t = 2.797

Confidence Interval:

So,

Answer is:

(3.724, 6.276)

The width of the confidence interval decreases as sample size increases.

(e)

H0: Null Hypothesis: = 6

HA: Alternative Hypothesis: < 6

n = 6

= 5

s = 2.2804

df = 6 - 1 = 5

= 0.05

From Table, critical values of t = 2.571

Test Statistic is given by:

Since calculated value of t = - 1.074 is greater than critical value of t= - 2.571, the difference is not significant. Fail to reject null hypothesis.

(f)

By Technology, p- value = 0.1659.

Since p value = 0.1659 is greater than = 0.05, the difference is not significant. Fail to reject null hypothesis.