Question

Please clearly state the step number next to the answers given, thank you!

A random sample of 12 supermarkets from Region 1 had mean sales of 72.4 with a standard deviation of 6.2. A random sample of 16 supermarkets from Region 2 had a mean sales of 78.3 with a standard deviation of 5.3. Does the test marketing reveal a difference in potential mean sales per market in Region 2? Let μ1 be the mean sales per market in Region 1 and μ2 be the mean sales per market in Region 2. Use a significance level of α=0.05 for the test. Assume that the population variances are not equal and that the two populations are normally distributed.

Step 1 of 4: State the null and alternative hypotheses for the test.
Step 2 of 4: Compute the value of the t test statistic. Round your answer to three decimal places.
Step 3 of 4: Determine the decision rule for rejecting the null hypothesis H0. Round your answer to three decimal places. (e.g. Reject H0 if t or |t| is < or > ___________)
Step 4 of 4: State the test's conclusion. (Reject or Fail to Reject)

Answer 1

The provided sample means are shown below:

Also, the provided sample standard deviations are:

and the sample sizes are

Step 1

Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

​

This corresponds to a two-tailed test, for which a t-test for two population means, with two independent samples, with unknown population standard deviations will be used.

Step 2

Test Statistics

Since it is assumed that the population variances are unequal, the t-statistic is computed as follows:

Step 3

Rejection Region

Based on the information provided, the significance level is α=0.05, and the degrees of freedom are df=21.603. In fact, the degrees of freedom are computed assuming that the population variances are unequal.

Hence, it is found that the critical value for this two-tailed test is tc​=2.076, for α=0.05 and df=21.603.

The rejection region for this two-tailed test is R={t:∣t∣>2.076}.

Step 4

Decision about the null hypothesis

Since it is observed that ∣t∣=2.649>tc​=2.076, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p=0.0148, and since p=0.0148<0.05, it is concluded that the null hypothesis is rejected. It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that population mean μ1​ is different than μ2​, at the 0.05 significance level.

Graphically

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