Paige:

Question 1.

You MUST assume the following :- (1) The original data are normally distributed with **same** variance (or Std Dev). (2) Each of the sample is a random sample. (3) The two samples are independently obtained. If any of the assumptions is not true, the conclusions MAY NOT BE VALID.

Step 1. Calculate the unbiased estimate of the common variance, and then the standard error.

The sample variance of Sample 1 (called S_{1}^{2}) is (given Sample SD)^{2} = (3.4)^{2} = 11.56

The sample variance of Sample 2 (called S_{2}^{2}) is (given Sample SD)^{2} = (5.6)^{2} = 31.36

(RED FLAG, the sample estimates of individual variances are VERY different, so the assumption that the population variances are the SAME may not be TRUE!)

Compute the estimate of the common variance = [(n_{1} - 1) S_{1}^{2} + (n_{2} -1) S_{2}^{2}] / (n_{1} + n_{2} - 2) = [(31-1)(11.56) + (32 -1)(31.36)] / (31+32-2) = 21.6223.

Std Error (Say SE) is **SE = sqrt (21.6223) = 4.65**.

Step 2. Compute the test statistic. t = [x_{1} bar - x_{2} (bar)] / {(SE) sqrt [(1 / n_{1}) + (1 / n_{2})]}

t = [29 - 35] /{4.65 [sqrt (1/31 + 1/32)]} = - 5.12 (if my arithmetic is correct)

Now set up the null and the alternative hypotheses.

Step 3. Null hyp: (mu)_{1} = (mu)_{2} , where (mu)_{1} and (mu)_{2} are the respective population means.

Since the question asks " is there a significant difference......" and the difference is not in one direction or the other, the alt. hyp is (mu)_{1} NOT EQUAL TO (mu)_{2}.

The alternative hypothesis is two-sided. So a two sided rejection is used. If the absolute value of the computed t-statistic is greater than an an appropriately chosen "critical value" (see below) for the given level of significance, then the data provide sufficient evidence against the null hypothesis at the given significance level, that is the "difference is significant" at the given significance level.

Step 4. Obtain the critical value.

Since both sample sizes are "large" (rule of thumb, each greater than 30), the critical value is obtained from the standard normal tables. The given sig. level is 0.01. Since the rejection region is two sided, the sig level is divided in half. 0.01/2 = 0.005. From the standard normal tables the "z_value" corresponding to 0.005 (such that the area under the standard normal curve to the right of the z_value is 0.005 is obtained. It comes out to be 2.576.

Step 5. Do the test.

Since the absolute value of the calculated t_statistic (Step 2) is |- 5.12| = 5.12 AND it is larger than 2.576, we say that the data provide sufficient evidence to reject the null hypothesis that the population means of females and males (for WHATEVER was measured, the problem does NOT state that) are the same and that there is a significant difference between the attitudes of males and females at 0.01 level of significance.

Paige, please follow and understand all the steps. Post a comment. Then I will help you with Question 2. There he sample sizes are SMALL, so a "t-table" has to be used in Step 4.

Dr. G.