Question

Assume that you have a sample of n 1 =4​, with the sample mean Upper X overbar1=46​, and a sample standard deviation of Upper S 1=5​, and you have an independent sample of n 2=7 from another population with a sample mean of Upper X overbar 2=37 and the sample standard deviation Upper S 2=6.

Assuming the population variances are​ equal, at the 0.01 level of​ significance, is there evidence that μ1>μ2​?

Determine the hypotheses. Choose the correct answer below.

Answer 1

Solution:-

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: u₁< u₂
Alternative hypothesis: u₁ > u₂

Note that these hypotheses constitute a one-tailed test.

Formulate an analysis plan. For this analysis, the significance level is 0.01. Using sample data, we will conduct a two-sample t-test of the null hypothesis.

Analyze sample data. Using sample data, we compute the standard error (SE), degrees of freedom (DF), and the t statistic test statistic (t).

SE = sqrt[(s₁²/n₁) + (s₂²/n₂)]
SE = 3.37533
DF = 9

t = [ (x₁ - x₂) - d ] / SE

t = 2.67

where s₁ is the standard deviation of sample 1, s₂ is the standard deviation of sample 2, n₁ is thesize of sample 1, n₂ is the size of sample 2, x₁ is the mean of sample 1, x₂ is the mean of sample 2, d is the hypothesized difference between population means, and SE is the standard error.

The observed difference in sample means produced a t statistic of 2.67.

Therefore, the P-value in this analysis is 0.013

Interpret results. Since the P-value (0.013) is greater than the significance level (0.01), we cannot reject the null hypothesis.

From the above test we do not have sufficient evidence in the favor of the claim that μ₁ > μ_2​.