Question

We are interested in understanding if there is a difference between the consumption of personal training between two of our tapestry segments. Segment 1 uses personal training 6.3 times on average per year (n=1000). Segment 2 uses personal training 5.8 times on average per year (n=850). The standard deviation for segment 1 is 2.3 and for segment 2 is 3.2. Is this a significant difference? Why or why not? Assume we are looking for 99% significance.

Answer 1

The provided sample means are shown below:

Also, the provided population standard deviations are:

and the sample sizes are = 1000 and = 850

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho:

Ha:

This corresponds to a two-tailed test, for which a z-test for two population means, with known population standard deviations will be used.

(2) Rejection Region

Based on the information provided, the significance level is α = 0.01, and the critical value for a two-tailed test is z_c = 2.58

(3) Test Statistics

The z-statistic is computed as follows:

z = 3.797

(4) Decision about the null hypothesis

Since it is observed that |z| = 3.797 > z_c = 2.58, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p = 0.0001, and since p = 0.0001 < 0.01, it is concluded that the null hypothesis is rejected.