Consider an upper-tail test of hypothesis for the difference in two population means. The sample data from the first population is: n = 50, x-bar = 110, s = 10. The sample data from the second population is: n = 75, x-bar = unknown, s = 11. We conclude the alternative hypothesis with a 2% level of significance. Then the value of the second sample mean must be:

Question

Consider an upper-tail test of hypothesis for the difference in two population means. The sample data from the first population is: n = 50, x-bar = 110, s = 10. The sample data from the second population is: n = 75, x-bar = unknown, s = 11. We conclude the alternative hypothesis with a 2% level of significance. Then the value of the second sample mean must be:

Answer 1

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

SEdiff = √(SEmean1^2 + SEmean2^2)

SEm = SD/√n

If only one SD is provided, you can use just that to determine SEdiff.

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (.02) for the Z score.

Answer 2

To find the value of the second sample mean, we need to perform an upper-tail test of hypothesis for the difference in two population means. Let's break down the steps to find the answer.

Step 1: State the null and alternative hypotheses.
The null hypothesis (H0) assumes that there is no difference between the means of the two populations, while the alternative hypothesis (Ha) assumes that there is a difference.

Since we want to conclude the alternative hypothesis, the first step is to state Ha. In this case, the alternative hypothesis for an upper-tail test would be:
Ha: μ1 - μ2 > 0 (where μ1 represents the mean of the first population and μ2 represents the mean of the second population).

Step 2: Set the significance level.
The question mentions a 2% level of significance, which means we want to test the hypothesis at a 2% significance level (α = 0.02).

Step 3: Formulate the decision rule.
In an upper-tail test, the critical region is located in the upper tail of the distribution. We need to find the critical value corresponding to the significance level.

To find the critical value, we can use a z-table or a statistical software. Assuming a normal distribution, for a 2% significance level (α = 0.02), the critical value is approximately 2.05 (corresponding to a z-score of 2.05).

Step 4: Calculate the test statistic.
The test statistic for comparing two population means is the difference of the sample means divided by the standard error.
In this case, the test statistic (z) can be calculated as follows:

z = ((x-bar1 - x-bar2) - 0) / √(s1^2 / n1 + s2^2 / n2)

Given the sample data for the first population:
n1 = 50
x-bar1 = 110
s1 = 10

Given the sample data for the second population:
n2 = 75
x-bar2 = unknown
s2 = 11

Since the value of the second sample mean (x-bar2) is unknown, we cannot calculate the test statistic without additional information.

Step 5: Make the decision.
Without the test statistic, we are unable to make a decision. We need the actual value of the second sample mean or more information to calculate the test statistic and compare it to the critical value.

In summary, based on the given information, we cannot determine the value of the second sample mean without additional data or the actual test statistic.