Hypothesis test (when to use z-test or t-test)

Question

I've got 6 questions here. I don't really need the complete answers, I just want to know what test (z or t) should be used and what are the basis for using that test. Here we go:

(1) According to the Department of Education, full-time graduate students receive an average salary of Php12,800. The dean of graduate studies at a large state university claims that his graduate students earn more than this. He surveys 46 randomly selected students and finds their average salary is Php13,445 with a standard deviation of Php1800. With alpha = 0.05, is the dean’s claim correct?

(2) A health researcher read that a 200-pound male can burn an average 0f 524 calories per hour playing tennis. 37 males were randomly selected and the mean number of calories burned per hour playing squash was 534.8 with a standard deviation of 45.9 calories. Do squash players burn more calories per hour than tennis players? Test with alpha = 0.01.

(3) To see if young men ages 8-17 years spend a different amount than the national average of Php 244.40 per shopping trip to a local mall, the manager surveyed 30 young men. She found that the average amount spent per trip was Php233.70 with a standard deviation of Php37.00. With alpha = 0.05, can it be concluded that 8-17 years old spend different amount at the local mall than the national average?

(4) A nutritionist believes that 12-ounce boxes of cereal should contain an average of 1.2 ounces of bran. She suspects that a popular cereal has a different mean bran content. She carefully analyzes the contents of a random sample of twenty 12-ounce boxes of the cereal and finds that the mean bran content is 1.16 ounces. It is known that the standard deviation of the bran content of all such boxes of cereal is 0.08.

(5) A famous nonverbal communication scholar believes that anxiety is an important variable in nonverbal behavior. On the basis of the assumption that anxious people have more energy that those who are calm, he hypothesizes that anxious people gesture more often while they talk than do calm people. He tests this hypothesis by randomly selecting five anxious people and five calm people (measured on a valid and reliable scale) from a university class. He tells each research participant to bring a friend to a laboratory where he videotapes 20 minutes of their conversation. He then counts the average of gestures per minute made by each research participant. He obtains the following scores: (a) Anxious people: 3,3,4,5,5; (b) Calm people: 4,6,7,9,9

(6) A clever undergraduate Grace, hypothesizes that people under 30 are more humorous than people over 30. She test this hypothesis by randomly selecting 4 people over 30 from the telephone book and 4 people under 30 from university registration lists. She gives each research participant the Humorous Speech Scale, which measures humor in a Likert-type scale from 1 to 10. The scores are listed below. (a) Under 30: 6,7,10,9; (b) Over 30: 5,6,2,3

Remember, I just need to figure out what test should be used. I can do the calculations. Please don't be rude. Thanks.

Answer 1

If $n$ is large enough (in general, $n \geq 30$, then you can use a $z$-test. If you don't know $\sigma$, you can still use $s$. The $t$-test is used when $n < 30$. Based on that information, you should be able to figure out four of these questions. However, two questions here are a little different. Depending on what you mean by "$z$-test" and "$t$-test" (to me they generally suggest determining something about a mean), you may have to use a different test, although $z$- or $t$-scores may be involved.

Let me know if you have questions.

Answer 2

There are several things to consider. First, there is a question about which test statistic to use. The z test statistic requires that you know the population standard deviation, which is implausible except when the data are drawn from a Bernoulli distribution.

The second question is what distribution to use for your critical values. The choices are normal and t. For large sample sizes (perhaps >30) it does not matter much since the normal and t distributions then almost coincide.

For small sample sizes it does matter. However, for small sample sizes either test only makes sense if the original data are approximately normally distributed, which is questionable in your examples.

In sum:

1. If you have a large sample size you can use the t-statistic using critical values from the normal (limit) distribution.

2. If your sample size is small and your data are normally distributed and you know the population standard deviation then you should use the z-statistic using critical values from the normal distribution.

3. If your sample size is small and your data are normally distributed and you don't know the population standard deviation then you should use the t-statistic using critical values from the t distribution.

4. In all other cases you should be looking for other, situation-specific, tests.

My take on this is that you rarely know the population standard deviation and you rarely have normally distributed data, so it's either case 1 or case 4.