STA218 Assignment for Quiz 10

Quiz on Thursday April 5th at the beginning of class

For this quiz, you will be allowed a single 8.5 by 11 one-sided formula sheet and a calculator. So write your formula sheet and use it to do the homework. We will supply a copy of the normal table. You might need the normal table for p-values.

Here is a set of standard questions that apply to any statistical test. These are the questions you will see on both the quiz and the final exam. Any reference to the "standard questions" means these:

1. State the statistical model you adopt for these data.
2. State the null hypothesis in symbols.
3. State the alternative hypothesis in symbols.
4. Calculate the test statistic; show your work. The answer is a number. Circle your answer.
5. Calculate the p-value (observed significance level). Show your work, including a picture of a curve with the tail area or areas shaded, and circle your final answer. Note: this applies only to Z-tests, not t-tests. You need a computer to get p-values for a t-test.
6. State the rejection region and/or draw a picture of it. Use α = 0.05. Some of the homework problems ask for different values, but you might as well practise using α = 0.05, because that's the traditional value and that's what I'm going to ask.
7. Do you reject the null hypothesis? Answer Yes or No.
8. Is p < α? Answer Yes or No.
9. Are the results statistically significant at the 0.05 level? Answer Yes or No.
10. Can we conclude …  (This part will depend on the question).

Some discussion may be useful before we start. The textbook frequently wants you to choose a one-sided test when it is clear to me that a two-sided test is better. Many of the examples and homework problems even provide a model of looking at the results first, and then phrasing the research question (corresponding to the alternative hypothesis) so that it implies a one-tailed test in the direction of the results that were obtained (Like "can we conclude that the engine is really using too much fuel?") If the results had come out in the other direction, they would have phrased the question differently, and adopted a rejection region in the opposite tail. Effectively, this is the same as doing a two-tailed test, but with a significance level that is twice what you are claiming.

This practice was so common in real applications of statistical tests, and so hard to eradicate, that after a brief period of "controversy," scientific journals mostly stopped publishing claims that were backed up with one-sided tests. Basically, our textbook is about 30 years behind the times in this respect, though it is modern in other ways.

There are a few cases, mostly in business or government and not in science, where a one-tailed test is still appropriate. If you can really be sure that there would be absolutely no consequences, and no action of any kind if the results came out opposite to what you are looking for, then go ahead and do a one-tailed test. I gave an example in class. If a factory is emitting poisonous chemicals above a certain (average) standard, then there are serious regulatory consequences. If they are below the standard, nobody cares; it's more or less expected.

The rule for our class is that you will always do two-sided tests unless you are explicitly told that the problem calls for a one-sided test. This means that the answers in the back of the book (as well as those scanned and posted by Lennon) will often be "wrong," at least in terms of the alternative hypotheses, p-values and conclusions. However, the numerical values of Z and t will still be okay.

Please read pages 271-279. You are responsible for: null hypothesis, alternative hypothesis, rejection region, critical value, Type I error, Type II error. The quantity α (introduced in the discussion of Type I error) is called the significance level; it is very important. You will not be asked anything about the acceptance region because some people think you should never accept the null hypothesis. You will not be asked anything about β or about power, because we are running out of time.

Z-test for a single mean: Read 280-285. You're not responsible for power.

t-test for a single mean: Read 286-289. Note that you will only apply this method if you believe the data come from a normal distribution. It is not a tool for small samples in general. Our course does not provide any solutions for the case of small samples from a distribution that is not normal.

Please do the following exercises.

1. Do Exercise 8.11. Answer all the standard questions above, except forget about the p-value. Please assume a normal distribution. The wording of the problem implies a one-tailed test, but you should do a two-tailed test. Can you conclude that the manager's stock selection differs from 19%?
2. Do Exercise 8.12, except answer the standard questions rather than a through f of the problem. Use α=0.05. Can one conclude that the company is averaging more than 4.8% profit?
3. Please answer the standard questions for Exercise 8.14, using α=0.05. It is reasonable to assume that the weight of lobster caught in a trap is a normally distributed random variable (really!). Can one conclude that the mean landings per trap decreased?

p-values: Please read Pages 293-296 and do Exercises 8.16 and 8.18. You are responsible for the concept of a p-value, and how to calculate it for Z-tests (not t-tests).

Z-test for two means: Please read pages 297-299. The phrase "when sample sizes are large" will mean n₁≥30 and n₂≥30, because the population variances will never be known; we'll always substitute sample variances in the formula for Z. Also, D₀, the difference between population means that is stated in the null hypothesis, will always be zero; you might as well record it that way (with no ) on your formula sheet.

t-test for two means (2-sample t-test, or independent t-test): Please read pages 299-304. There are several things to note here.

• In the panel on page 300, the symbol s² has a different meaning than it does in the rest of the book. It is not the sample variance. Rather, based on the assumption that the two samples come from populations with possibly different means but a common variance σ², it is a pooled estimate of σ² based on the two sample variances.
• For us, D₀ will always equal zero.
• In spite of what the book says on page 303, the normal distribution assumption does matter for the two-sample t-test when the sample sizes are small. It's when the sample sizes are both large that normality does not matter much. The book is correct that the equal variance assumption matters. However, it matters a lot less when the sample sizes are large and nearly equal. All this is "academic," because our table of the t distribution does not include critical values for large samples.

  The practical implication of all this for you is that you will only apply the t-test when the problem explicitly states that the distribution is normal. As in the case of the one-sample test, this course does not provide tools for small samples from non-normal populations.

• Using the t-table, it is possible to give inequalities about the p-values for t-tests, like 0.10 < p < 0.20. You will not be asked to do this.
• You are not responsible for the "t-test" that applies when the population variances are unequal. The formula for the degrees of freedom is too hairy.

Now please do these exercises.

1. Use the data from Exercise 8.32 to test for a difference between means, but please stick to our standard questions instead of the book's questions, using α=0.05 instead of α=0.10. Is there evidence of a change in approved home loan applications?
2. Use the data from Exercise 8.33 to test for a difference between means, but please stick to our standard questions instead of the book's questions. Assume a normal distribution and equal variances. Did the programme improve customer satisfaction?

Paired difference tests: When the data consist of differences between matched pairs of data, like before-after or a comparison of wheels on the same automobile, you just apply a one-sample test to the differences, usually testing H₀: μ=0. The book presents this as a separate method (called a paired-difference test) and presents separate formulas with d-bar instead of x-bar and so on, but it's just the same old formulas with a slightly more obscure notation. The book presents only t-tests, but of course you could do a one-sample Z-test on differences from a large sample from a possibly non-normal distribution -- of differences. Note that for the paired-difference t-test, it is assumed that the differences are normally distributed. Anyway, please read pages 307-312. Their formula for a confidence interval is nothing new.

Look at exercise 8.43. This really is a dog, no, that's unfair to dogs. The assumption of normality is extremely questionable. Don't do that problem. Instead, do 8.44, answering the standard questions. Assume normality, which they forgot to say, again. Don't bother with the confidence interval.

Z-test for a single proportion: Please read pages 316-318. Note the useful and clear discussion (p. 317, repeated in blue on p. 318) of why you use p₀ instead of p-hat in the standard error. And as usual, their x, in p-hat=x/n, is our Σx_i, where the data are binary.

Do Exercise 8.52, sticking to our standard questions instead of the book's questions. First test if the sample size is big enough (use p₀). Can you conclude that the check verification programme reduced the number of bad checks?

Z-test for difference between proportions: Please read pages 320-323. As usual, D₀ will always equal zero, so only the second-to-last formula for Z on page 321 is needed. The formula for "p-hat" reduces to the proportion of ones in the two samples combined. That's right; merge the two samples into one big sample, and calculate the sample proportion. You will not be responsible for checking whether the sample sizes are big enough.

Please do Exercise 8.58, but answer the standard questions. Ignore the questionable suggestion that the modification of the production line "could not possibly increase the fraction defective." What would you do if it did? As usual, we do a t-tailed test.

You are not responsible for any tests or confidence intervals for variances.

Do the assigned problems in preparation for the quiz. They are not to be handed in.