Now, suppose we drew a sample of 100 people from this course and obtained a mean score of 530. Which answer below best reflects the null hypothesis, *H*_{0}; alternative hypothesis, *H*_{1}; and conclusion based on an alpha of .05? Recall, the average score for people who took the VAST quantitative section with no training is 500 with a standard deviation of 100. Again, you may use a *z*-table or the *p*–*z* converter.

#### H_{0}: m = 530; H_{1}: m = 500; Do not reject null hypothesis.

*This answer is incorrect!* There are two problems. First, the correct null hypothesis should represent our initial assumption that there is no effect. In this case, the assumption is that the population of graduates of the training program has the same mean as the population of students who received no training (500). Second, the alternative hypothesis is incorrect. An alternative hypothesis includes every possibility other than the null hypothesis, rather than one specific value.

#### H_{0}: m ≤ 500; H_{1}: m = 530; Do not reject null hypothesis.

*This answer is incorrect!* Your alternative hypothesis is incorrect. An alternative hypothesis includes every possibility other than the null hypothesis, rather than one specific value.

#### H_{0}: m = 500; H_{1}: m ≠ 500; Do not reject the null hypothesis.

*This answer is incorrect!* Your hypotheses would be correct if you were conducting a two-tailed test. Moreover, you have drawn the wrong conclusion. Remember, we reject the null hypothesis when the observed result (sample mean) would have occurred less than 5% of the time given that the null is true. You may want to recheck your *z* calculation and *p*-values.

#### H_{0}: m ≤ 500; H_{1}: m > 500; Reject the null hypothesis.

**This answer is correct!** Your hypotheses are correct and you have drawn the correct conclusion. The probability of obtaining a *z*-score of 3.00 or greater given that the null is true is *p* = .001 or 0.1%. Therefore, we reject the null hypothesis.

Notice how the sample size affects the calculation of *z*-scores. For a sample size of 100, a sample mean of 530 corresponds to:

whereas, for a sample size of 10, the same sample mean corresponds to:

a *z*-score which does not allow us to reject the null hypothesis.