Question

Test the claim about the difference between two population means mu_1 and mu_2 at the level of significance alpha. Assume the samples are random and independent, and the populations are normally distributed. Claim: mu_1 = mu_2: alpha = 0.01 Population statistics: sigma_1 = 3.2, sigma_2 = 1.7 Sample statistics x_1 = 16, n_1 = 30, x_2 = 18, n_2 = 27 Determine the alternative hypothesis. H_a: mu_1 mu_2 Determine the standardized test statistic. z = (Round to two decimal places as needed.) Determine the P-value. P-value = (Round to three decimal places as needed.) What is the proper decision?
A. Fail to reject H_0. There is enough evidence at the 1% level of significance to reject the claim
B. Fail to reject H_0. There is not enough evidence at the 1 % level of significance to reject the claim.
C. Reject H_0. There is enough evidence at the 1% level of significance to reject the claim.
D. Reject H_0. There is not enough evidence at the 1% level of significance to reject the claim.

Answer 1

Answer:

a) Alternative hypothesis, $H_{a} : \mu_{1} \neq \mu_{2}$

b) z = -2.91, Pvalue = 0.002

c)Option C. Reject H_0. There is enough evidence at the 1% level of significance to reject the claim.

Step-by-step explanation:

a) Null hypothesis, $H_{0} : \mu_{1} = \mu_{2}$

Alternative hypothesis, $H_{a} : \mu_{1} \neq \mu_{2}$

b) Standardized test statistic

Level of significance, $\alpha = 0.01$

$\sigma_{1} = 3.2$

$\sigma_{2} = 1.7$

$X_{1} = 16\\X_{2} = 18\\n_{1} = 30\\n_{2} = 27$

$z = \frac{\mu_{1}- \mu_{2} }{\sqrt{\frac{\sigma_{1} ^{2} }{n_{1} } + \frac{\sigma_{2} ^{2} }{n_{2} } }}$

$z = \frac{16-18 }{\sqrt{\frac{3.2^{2} }{30 } + \frac{1.7 ^{2} }{27 } }}$

z = -2.91

Checking the p-value that corresponds to z = -2.906

P-value = 0.002

c) What is the proper decision

P-value  = 0.002

Level of significance, α = 0.01

0.002 < 0.01

Since Pvalue < α, the null hypothesis H₀ should be rejected.

Option C is the correct option.