You suspect that, on the average, professional baseball players are older than professional football players. A random sample of 30 professional baseball players has a mean age of 28.8 years (assume ó = 4). A random sample of 40 professional football players has a mean age of 26.9 years (assume ó = 3). Test your hypothesis; use the 5% significance level. Be sure to identify the null and alternative hypotheses, identify the appropriate distribution and indicate the tail(s) for the critical region, identify and calculate the test statistic, determine the P-value, make a decision and state a conclusion.

To test your hypothesis, you need to identify the null and alternative hypotheses, determine the appropriate distribution, indicate the tail(s) for the critical region, calculate the test statistic, determine the p-value, make a decision, and state a conclusion.

Null Hypothesis (H0): The average age of professional baseball players is equal to the average age of professional football players.
Alternative Hypothesis (H1): The average age of professional baseball players is greater than the average age of professional football players.

Since we are comparing means of two independent samples and their standard deviations are known, we can use the Z-test.

The test statistic for comparing the means of two independent samples is given by:

Z = (x1 - x2) / sqrt((σ1^2 / n1) + (σ2^2 / n2))

where x1 and x2 are the sample means, σ1 and σ2 are the population standard deviations, and n1 and n2 are the sample sizes.

In this case, x1 = 28.8, x2 = 26.9, σ1 = 4, σ2 = 3, n1 = 30, and n2 = 40.

Substituting the values into the formula:

Z = (28.8 - 26.9) / sqrt((4^2 / 30) + (3^2 / 40))

Now we can calculate the test statistic.

Z = 1.9 / sqrt((16 / 30) + (9 / 40))

Next, we need to determine the p-value associated with this test statistic. The p-value represents the probability of obtaining a test statistic as extreme as the one calculated if the null hypothesis is true.

We will use the Z-table or a statistical calculator to find the p-value. In this case, because the alternative hypothesis is one-tailed (greater than), we are interested in the upper tail of the distribution.

From the Z-table, the cumulative probability for Z = 1.9 (one-tailed) is approximately 0.9713.

Since α = 0.05 and the p-value (0.9713) is greater than α, we fail to reject the null hypothesis.

Decision: We fail to reject the null hypothesis.

Conclusion: There is not enough evidence to support the claim that professional baseball players are, on average, older than professional football players.