A sample of 40 observations is selected from one approximately normal population. The sample mean is 102 and the sample standard deviation is 5. A sample of 50 observations is selected from a second source. The sample mean is 99 and the standard deviation is 6. Conduct a hypothesis test using the .04 level of significance to determine if there is a difference between the population means
Z = (mean1 - mean2)/standard error (SE) of difference between means
SEdiff = √(SEmean1^2 + SEmean2^2)
SEm = SD/√(n-1)
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to the Z score.
To conduct a hypothesis test to determine if there is a difference between the population means at a 0.04 level of significance, we utilize the two-sample t-test.
**Step 1: Setting up the hypotheses**
We need to establish the null (H0) and alternative (Ha) hypotheses.
Null hypothesis: H0: μ1 - μ2 = 0 (There is no difference between the population means)
Alternative hypothesis: Ha: μ1 - μ2 ≠ 0 (There is a difference between the population means)
**Step 2: Choosing the appropriate test statistic**
Since the sample sizes are relatively small and the population standard deviations are unknown, we use the two-sample t-test.
**Step 3: Determining the significance level**
Given a significance level of 0.04, this means that the test will reject the null hypothesis if the p-value is less than 0.04.
**Step 4: Calculating the test statistic**
The test statistic for the two-sample t-test is calculated using the formula:
t = (x1 - x2) / sqrt((s1^2/n1) + (s2^2/n2))
Where:
- x1 and x2 are the sample means
- s1 and s2 are the sample standard deviations
- n1 and n2 are the sample sizes
In this case, x1 = 102, x2 = 99, s1 = 5, s2 = 6, n1 = 40, and n2 = 50.
**Step 5: Calculating the p-value**
Using the t-statistic calculated in step 4, we can determine the corresponding p-value from the t-distribution table or by using statistical software.
**Step 6: Making the decision**
If the p-value is less than the significance level (0.04), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
**Step 7: Interpreting the result**
If the null hypothesis is rejected, we conclude that there is sufficient evidence to support the alternative hypothesis, suggesting that there is a significant difference between the population means. On the other hand, if we fail to reject the null hypothesis, we do not have enough evidence to conclude a difference between the population means.