Question

You know that y is a normally distributed variable with a variance of 9. You do not know its mean. You collect some data. For each sample below, form the 95% confidence interval and test the null hypothesis of the mean equaling 2.

a. (8,1,5)

b. (8,1,5,-4,-8)

c. (8,1,5,-4,-8,4,8,5)

Answer 1

We know the population Variance, ² = 9

So, Standard Deviation, = 3

We need to find 95% confidence interval in each case,

z-score = 1.96

Null Hypotheses, Ho: = 2

a.) Sample -> (8,1,5)

So, sample mean, m = 14 / 3 = 4.67

n = 3

Confidence Interval, CI = m ( z * / )

CI = 4.67 ( 1.96 * 3 / )

CI = 4.67 ( 3.394)

CI = (1.276, 8.064)

Since Confidence interval contains value 2, we cannot reject Null Hypotheses

b.) Sample -> (8,1,5,-4,-8)

sample mean, m = 2 / 5 = 0.4

n = 5

Confidence Interval, CI = m ( z * / )

CI = 0.4 ( 1.96 * 3 / )

CI = 0.4 ( 2.629)

CI = (-2.229, 3.029)

Since Confidence interval contains value 2, we cannot reject Null Hypotheses

C.) Sample -> (8,1,5,-4,-8,4,8,5)

sample mean, m = 19 / 8 = 2.375

n = 8

Confidence Interval, CI = m ( z * / )

CI = 2.375 ( 1.96 * 3 / )

CI = 2.375 ( 2.0789)

CI = (0.2961, 4.453)

Since Confidence interval contains value 2, we cannot reject Null Hypotheses