Questions: Which one of the following is an assumption made when using a two sample test of means with equal population standard deviations? The difference of sample means follows a normal distribution. The sampled populations are independent. The sample taken from each population must be at least 5.

Which one of the following is an assumption made when using a two sample test of means with equal population standard deviations? The difference of sample means follows a normal distribution. The sampled populations are independent. The sample taken from each population must be at least 5.
Transcript text: Which one of the following is an assumption made when using a two sample test of means with equal population standard deviations? The difference of sample means follows a normal distribution. The sampled populations are independent. The sample taken from each population must be at least 5.
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Solution

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Solution Steps

Step 1: Standard Error Calculation

The standard error (SE) for the difference of means is calculated using the pooled variance \( s_p^2 \) and the sample sizes \( n_1 \) and \( n_2 \):

\[ SE = \sqrt{s_p^2 \left(\frac{1}{n_1} + \frac{1}{n_2}\right)} = 0.2408 \]

Step 2: Test Statistic Calculation

The test statistic \( t \) is computed as follows:

\[ t = \frac{\bar{x}_1 - \bar{x}_2}{SE} = 0.5813 \]

Step 3: Degrees of Freedom

The degrees of freedom \( df \) for the test is given by:

\[ df = n_1 + n_2 - 2 = 5 + 5 - 2 = 8 \]

Step 4: P-value Calculation

The p-value \( P \) is calculated using the t-distribution:

\[ P = 2(1 - T(|t|)) = 2(1 - T(0.5813)) = 0.577 \]

Step 5: Critical Value

The critical value for a two-tailed test at a significance level of \( \alpha = 0.05 \) with \( df = 8 \) is:

\[ \text{Critical Value} = 2.306 \]

Final Answer

The results of the two-sample t-test are as follows:

  • Test Statistic: \( t = 0.5813 \)
  • Degrees of Freedom: \( df = 8 \)
  • P-value: \( P = 0.577 \)
  • Critical Value: \( 2.306 \)

Thus, the answer is boxed as follows:

\[ \boxed{t = 0.5813, \, df = 8, \, P = 0.577, \, \text{Critical Value} = 2.306} \]

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