In a simple random sample of 250 father-son pairs taken from a large population of such pairs, the mean height of the fathers is 68.5 inches and the SD is 2.5 inches; the mean height of the sons is 69 inches and the SD is 3 inches; the correlation between the heights of the fathers and sons is 0.5.

In the population, are the sons taller than their fathers, on average? Or is this just chance variation?
The SE of the mean difference between heights of fathers and sons in the sample is closest to 0.176.

Problem 1.
Which of the following most closely represents the result of the test?
- The result is not statistically significant, so we conclude that it is due to chance variation.
- The result is not statistically significant, so we conclude that the sons are taller than their fathers, on average.
- The result is highly statistically significant, so we conclude that the sons are taller than their fathers, on average.
- The result is highly statistically significant, so we conclude that it is due to chance variation

Z = (mean1 - mean2)/standard error (SE) of difference between means

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score. Are you using two-tailed test or one-tailed test?

You can eliminate some choices: significance = not chance and non-significance = chance.

What criterion are you using to reject your null hypothesis? P≤.05? P≤.01?

0.5

Building on above answer from PsyDAG, when you run the calculation for Z and check the normal distribution table, you get get a p value of 0.22%, which is highly statistically significant, so we conclude that sons are taller, on average, than their dads.

Being P so small, it must be the other answer?? right???

To determine whether the sons are taller than their fathers, on average, we need to conduct a statistical test.

The given information provides us with the following:
- Mean height of fathers: 68.5 inches with a standard deviation (SD) of 2.5 inches
- Mean height of sons: 69 inches with a SD of 3 inches
- Correlation between the heights of fathers and sons: 0.5

To compare the means of two independent samples (fathers and sons), we can use a t-test. However, since we only have information about a single sample (250 father-son pairs), we need to calculate the standard error (SE) of the mean difference between the heights of fathers and sons in the sample.

The formula to calculate the SE of the mean difference is:

SE = sqrt[(SD1^2/n1) + (SD2^2/n2)]

Where SD1 and SD2 are the standard deviations of the two samples, and n1 and n2 are the sample sizes.

In this case, the SE of the mean difference is closest to 0.176.

Once we have the SE, we can conduct a hypothesis test. The null hypothesis (H0) is that there is no difference between the average heights of fathers and sons. The alternative hypothesis (Ha) is that the sons are taller than their fathers, on average (i.e., one-sided alternative hypothesis).

Using the t-test, we can calculate the t-statistic by dividing the observed mean difference (69 inches - 68.5 inches) by the standard error (0.176).

If the calculated t-statistic falls within the critical region (determined based on the desired significance level and degrees of freedom), we reject the null hypothesis and conclude that the sons are taller than their fathers, on average. Otherwise, if the calculated t-statistic does not fall within the critical region, we fail to reject the null hypothesis and conclude that the difference between the average heights is due to chance variation.

Given only the information about the SE of the mean difference (0.176), we cannot determine the precise result of the test. We need to know the actual t-statistic value and compare it to the critical region to make a conclusive statement.

Therefore, the correct answer to problem 1 is:
- The result is not statistically significant, so we cannot draw a conclusion from the given information.