Question

You wish to test the following claim (Ha) at a significance level of α=0.005.

      Ho:μ1=μ2
      Ha:μ1>μ2

You believe both populations are normally distributed, but you do not know the standard deviations for either. However, you also have no reason to believe the variances of the two populations are not equal. You obtain a sample of size n1=15 with a mean of M1=88.3 and a standard deviation of SD1=7.1 from the first population. You obtain a sample of size n2=27 with a mean of M2=78.6 and a standard deviation of SD2=15.3 from the second population.

What is the critical value for this test? For this calculation, use the conservative under-estimate for the degrees of freedom as mentioned in the textbook. (Report answer accurate to three decimal places.)
critical value =

What is the test statistic for this sample? (Report answer accurate to three decimal places.)
test statistic =

The test statistic is...

• in the critical region
• not in the critical region

This test statistic leads to a decision to...

• reject the null
• accept the null
• fail to reject the null

As such, the final conclusion is that...

• There is sufficient evidence to warrant rejection of the claim that the first population mean is greater than the second population mean.
• There is not sufficient evidence to warrant rejection of the claim that the first population mean is greater than the second population mean.
• The sample data support the claim that the first population mean is greater than the second population mean.
• There is not sufficient sample evidence to support the claim that the first population mean is greater than the second population mean.

Answer 1

Critical value is the value after which we are going to reject the null hypothesis.

This is a one tailed test because our alternate hypothesis is μ1>μ2.

For α=0.005, using the smaller of the n-1 values for two samples (conservative under-estimate) as degree of freedom, we can calculate the critical value i.e. the t^* value.

df = 15-1 = 14 and α=0.005 and for one right tailed disribution

t^* = 2.977

Test statistic

t = (x₁ - x₂₎ / (s₁²/n₁ + s₂²/n₂)^1/2

where x₁ and x₂ are the two sample means s₁ and s₂ are sample standard deviation

n₁ and n₂ are the two sample sizes.

t = 2.797

We can see the test statistic is not in the critical region.

which leads to a decision to fail to reject the null hypothesis.

The final conclusion being

There is not sufficient sample evidence to support the claim that the first population mean is greater than the second population mean.