Question

For the given​ data, (a) find the test​ statistic, (b) find the standardized test​ statistic, (c) decide whether the standardized test statistic is in the rejection​ region, and​ (d) decide whether you should reject or fail to reject the null hypothesis. The samples are random and independent. ​

Claim: μ1 < μ2​, α = 0.01. Sample​ statistics: x^-1 = 1240, n1 = 40, x^-2 = 1200 and n2 = 80. Population​ statistics: σ1 = 65 and σ2 = 110.
(a) The test statistic for μ1 - μ2 is _________.
(b) The standardized test statistic for μ1 - μ2 is __________. (Round to two decimal places as needed.)
(c) Is the standardized test statistic in the rejection region?
O Yes Ο Nο
(d) Should you reject or fail to reject the null hypothesis?

Answer 1

Complete Question

The complete question is shown on the first uploaded image

Answer:

a

The test statistic for μ1 - μ2 is
`\= x_1 - \= x_2 = 40`

b

The standardized test statistic for μ1 - μ2 is
`z = 2.5`

c

No

d

Fail to reject null hypothesis
`H_O : \mu_1 \ge \mu_2 ; H_a : \mu_1 < \mu_2` At the 1% significance level , there is insufficient evidence to support the claim

Explanation:

From the question we are told that

The given data is

`\= x_1 = 1240`

`\= x_2 = 1200`

`n_1 = 40`

`\alpha = 0.01.`

`n_2 = 80.`

`\sigma 1 = 65 \ and\ \sigma2 = 110.`

Now the test statistic is mathematically evaluated as

`\= x_1 - \= x_2 = 1240 -1200`

`\= x_1 - \= x_2 = 40`

The standardized test​ statistic is mathematically represented as

`z = \frac{\= x_1 - \= x_2}{\sqrt{(\sigma_1^2)/(n_1) } + (\sigma_2^2)/(n_2) }`

substituting values

`z = \frac{\= 1240 - \= 1200}{\sqrt{(65^2)/(40) } + (110^2)/(80) }`

`z = 2.5`

Now the standardized test​ statistic is not in the rejection region because the z value of
`\alpha` is 2.33 and the standardized test​ statistic is greater than that hence it is not in the rejection region

This implies that the test statisties failed to reject the null hypothesis at significance level of 0.01 , there insufficient evidence to support the claim