In two independent samples from populations that are normally distributed, x¯1 =35.0, s1 = 5.8, n1 =12 and x¯2 = 42.5, s2 = 9.3, n2 = 14. Using the 0.05 level of significance, test H0: µ1 = µ2 versus H1: µ1 ≠ µ2

Here's how I would proceed.

First, we have a case of two independent unpaired samples from normally distributed populations, where
1=35
s1 = 5.8
n1=12
2=42.5
s2 = 9.3
n2=14

H: μ1 = μ2.
We calculate the t-statistic
t = (x̄2 - x̄1)/sqrt(s1²/n1+s2²/n2)
=(42.5-35)/sqrt(5.8²/12+9.3²/14)
=7.5/2.99686
=2.503
The number of degree of freedoms is given by:
dof
=((s1²/n1) + (s2²/n2))²/((s1²/n1)²/n1 + (s2²/n2)²/n2)
=(5.8²/12+9.3²/14)²/((5.8²/12)²/11 + (9.3²/14)²/13)
=80.662/3.65
= 22.1

The two-tail t-statistic of 2.503 and a degree of freedom of 22 has a probability of 0.020, less than the significance level of 5%, so the hypothesis is rejected.

A similar calculation applies to the hypothesis of μ1 ≠ μ2.

Please check my calculations.

There is a very good article on the student's t-test, and an interactive t-statistic table.
http://en.wikipedia.org/wiki/Student's_t-test
http://www.tutor-homework.com/statistics_tables/statistics_tables.html#t

To test the hypothesis H0: µ1 = µ2 versus H1: µ1 ≠ µ2, we can use the two-sample independent t-test. This test compares the means of two independent samples to determine if they are significantly different from each other.

The steps to perform the test are as follows:

1. State the null and alternative hypotheses:
- Null hypothesis (H0): The means of the two populations are equal (µ1 = µ2).
- Alternative hypothesis (H1): The means of the two populations are not equal (µ1 ≠ µ2).

2. Determine the significance level (α):
- In this case, the level of significance is given as 0.05.

3. Compute the test statistic:
- The formula for the test statistic is:
t = (x¯1 - x¯2) / √((s1^2/n1) + (s2^2/n2))
- Plug in the given values:
t = (35.0 - 42.5) / √((5.8^2/12) + (9.3^2/14))

4. Determine the critical value:
- The critical value(s) depend on the significance level and the degrees of freedom.
- Degrees of freedom (df) = (n1 - 1) + (n2 - 1) = 11 + 13 = 24
- Since this is a two-tailed test (µ1 ≠ µ2), we need to find the critical values for α/2 = 0.05/2 = 0.025.
- Looking up the critical values in a t-table for df = 24 and α/2 = 0.025, we find t-critical = ±2.0639.

5. Make a decision:
- If the test statistic falls in the rejection region (outside the t-critical values), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
- In this case, compare the computed test statistic to the critical values. If |t| > t-critical, then reject H0. Otherwise, fail to reject H0.

6. Calculate the p-value (optional):
- The p-value represents the probability of obtaining a test statistic as extreme as the observed value, assuming the null hypothesis is true.
- For a two-tailed test, we calculate the p-value by finding the area in both tails of the t-distribution beyond the test statistic.
- We can use a t-table or statistical software to find the p-value.

So, to conclude, compare the computed test statistic to the critical values (±2.0639) and make a decision about the null hypothesis. Additionally, you can calculate the p-value to determine the statistical significance of the results.