**Answer:**

**Step-by-step explanation:**

Data given:

Today Five Years Ago

Mean 82 88

Variance 112.5 54

Sample Size 45 36

**Data given and notation
**

represent the mean for the sample today

represent the mean for the sample five years ago

represent the population standard deviation for the sample today

represent the population standard deviation for the sample five years ago

sample size for the group today

sample size for the group five years ago

t would represent the statistic (variable of interest)

significance level

**Develop the null and alternative hypotheses for this study?
**

We need to conduct a hypothesis in order to check if the means for the two groups are different, the system of hypothesis would be:

Null hypothesis:

Alternative hypothesis:

Since we know the population deviations for each group, for this case is better apply a z test to compare means, and the statistic is given by:

(1)

**z-test:** Is used to compare group means. Is one of the most common tests and is used to determine whether the means of two groups are equal to each other.

**Determine the critical value(s).
**

Based on the significance level assumed and we can find the critical values with the normal standard distribution, we are looking for values that accumulates 0.005 of the area on each tail on the normal distribution.

For this case the two values are and

**Calculate the value of the test statistic for this hypothesis testing.
**

Since we have all the values we can replace in formula (1) like this:

**What is the p-value for this hypothesis test?
**

Since is a bilateral test the p value would be: