Consider the following hypothesis statement using alpha = .05, and data from two independent samples:

Ho: µ1 - µ2 = 0
Ha: µ1 - µ2 not equal to 0

X1 = 237, X2 = 218
¬1 = 54, ¬2 = 63
n1 = 42, n2 = 35

a) Calculate the appropriate test statistic and interpret the result.
b) Calculate the p value and interpret the result.

Z = (mean1 - mean2)/standard error (SE) of difference between means

SEdiff = √(SEmean1^2 + SEmean2^2)

SEm = SD/√n

If only one SD is provided, you can use just that to determine SEdiff.

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score.

To calculate the appropriate test statistic and interpret the result, we need to follow these steps:

Step 1: Calculate the difference in sample means (X1 - X2)
We have X1 = 237 and X2 = 218. So the difference in sample means is (237 - 218) = 19.

Step 2: Calculate the pooled standard error (s.e.)
To calculate the pooled standard error, we need to calculate the variances of the two samples and then take the square root of the sum of the variances divided by the total sample size.

For sample 1:
n1 = 42
¬1 = 54

The variance of sample 1 can be calculated by dividing the sum of squared differences from the mean by (n1 - 1):
Var1 = [ (∑(¬1 - X1)^2) / (n1 - 1) ] = [ (∑(54 - 237)^2) / (42 - 1) ]

For sample 2:
n2 = 35
¬2 = 63

Similarly, calculate the variance of sample 2:
Var2 = [ (∑(¬2 - X2)^2) / (n2 - 1) ] = [ (∑(63 - 218)^2) / (35 - 1) ]

The pool standard error is then calculated as:
s.e. = sqrt[ (Var1 / n1) + (Var2 / n2) ]

Step 3: Calculate the test statistic
To calculate the test statistic, we divide the difference in sample means (Step 1) by the pooled standard error (Step 2):
Test Statistic = (X1 - X2) / s.e.

Step 4: Interpret the result
The test statistic follows a t-distribution with degrees of freedom equal to the sum of the sample sizes minus 2 (df = n1 + n2 - 2). We can compare the test statistic to the critical value from the t-distribution table to determine if there is enough evidence to reject the null hypothesis.

If the calculated test statistic is greater than the critical value, then we reject the null hypothesis. If it is smaller than the critical value, we fail to reject the null hypothesis.

To calculate the p-value and interpret the result, we need to follow these steps:

Step 5: Calculate the p-value
The p-value is the probability of getting a test statistic as extreme as the one calculated (or more extreme) under the null hypothesis.

To calculate the p-value, we can use statistical software or t-distribution tables. We compare the absolute value of the test statistic to the critical value in the t-distribution table and find the corresponding p-value.

Step 6: Interpret the result
If the p-value is less than the chosen significance level (alpha), which is 0.05 in this case, then we reject the null hypothesis. If the p-value is greater than the chosen significance level, we fail to reject the null hypothesis.

Remember, rejecting the null hypothesis implies that there is enough evidence to suggest that there is a significant difference between the two means (µ1 - µ2 is not equal to 0), while failing to reject the null hypothesis implies that there is not enough evidence to suggest such a difference.