Question

A study was done using a treatment group and a placebo group. The results are shown in the table. Assume that the two samples are independent simple random samples selected from normally distributed​ populations, and do not assume that the population standard deviations are equal. Complete parts​ (a) through​ (c) below. Use a 0.05 significance level for both parts.

	Treatment	Placebo
μ	μ1		μ2
n	27		31
x overbarx	2.31		2.62
s	0.99		0.67

a. Test the claim that the two samples are from populations with the same mean. What are the null and alternative​ hypotheses?

b. What is the Test Statistic, t? What is the P-value?

State the conclusion for the test.

c. Construct a confidence interval suitable for testing the claim that the two samples are from populations with the same mean.

Answer 1

Since we are considering unequal variances, we calculate the degrees of freedom as given below.

The Hypothesis:

H0: =

Ha:

This is a Two tailed test.

(b) The Test Statistic:

The p Value:    The p value (2 Tail) for t = -1.38, df = 45, is; p value = 0.0872

The Decision Rule: If the P value is < , Then Reject H0

The Decision: Since P value (0.0872) is > (0.05), We Fail to Reject H0.

The Conclusion: There isn't sufficient evidence at the 95% significance level to warrant rejection of the claim that the 2 means are from the same population.

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(c) Given:

= 2.31, s₁ = 0.99, n₁ = 27,

= 2.62, s₂ = 0.67, n₂ = 31,

= 0.05, df = 45

The tcritical (2 tail) = 2.014

The Confidence Interval is given by (- ) ME, where

(- ) = 2.31 – 2.62 = -0.31

The Lower Limit = -0.31 - 0.45 = -0.76

The Upper Limit = -0.31 + 0.45 = 0.14

The 95% Confidence Interval is (-0.76 , 0.14)

Here again we see, that the limits have 2 different signs, and hence the confidence interval contain the value 0, which means it is a possibility that the 2 means are equal, and therefore we cannot reject the null hypothesis.

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