More help with Chapters 14 and 15.

Chapter 14.

You have learned to do the basics of confidence intervals and hypothesis tests. Now we work on understanding these more deeply.

1. What does it mean to have 95% confidence? Answer: In "Margin of Error and Confidence Level" page 363, explain what the figure means. What is it telling you? See exercise 14.2 and do more of these yourself. It is very important that you understand this picture and it comes into your mind whenever you are thinking about a confidence interval.
   
   How does that relate to the way we write the interpretation of the confidence interval, "I have 95% confidence that the population mean ..... lies between ... and ...." ? Answer: The confidence interval is about the population mean, and it is not 100% confidence. It is not about the sample mean, nor is it about individual observations from the population.
   
   
2. "Simple conditions" on page 367, at the end of "Confidence Intervals for a Population Mean."
   
   This same set of conditions applies to Significance Test problems in this chapter as well, because they are the conditions needed to use the sampling distribution of x-bar as we use it to form confidence intervals and do hypothesis tests.
   
   The "conditions for inference" are a major focus of Tests 4 and 5. You are being introduced to the idea gradually. It is important that you pay attention to it as you go along. Ask questions as needed!
   
   
3. Stating hypotheses.
   • Every Ho has an equal in it.
   • The Ho and Ha never overlap.
   • Generally, what is in the Ha is something that will make us want to change something - go to some trouble, etc. So we need reasonably strong evidence to do that. That supports the way we do hypothesis tests - usually with fairly small significance levels.
   • Always write your hypotheses and select your significance level before you look at the data. The logic and mathematical underpinings of probability require that in order for the conclusions to be valid.
     
     
4. Definition and meaning of the p-value: The probability of seeing data
   • at least as far away from the Ho value as our data
   • in the direction of Ha
   • if the Ho is true.
     
     
5. Why does a two-sided alternative hypothesis make us multiply the area by 2 to find the p-value? Answer: Because "in the direction of the Ha" is now two directions, and there is an area at least as far away as our data in both of those directions.
   
   
6. Remember the pictures you drew for exercises 14.6 and 14.7 to illustrate which types of sample means give strong evidence against Ho. Use that to remind yourself of why small p-values lead you to reject Ho and not-so-small p-values make us choose not to reject Ho.
   
   
7. How small a p-value? We choose a significance level to express how small the p-value needs to be. Choose among 1%, 5%, or 10%. You may or may not be given the significance level in the problem. Choose it before you look at the data.
   
   
8. What is "statistical significance"? A set of data is "statistically signficant" if it leads to a small enough p-value to make you reject the Ho. (Recall that, if you believe Ha, then you will want to "change something" or "spend money." So we want to see significant evidence before we go to that trouble.
   
   
9. "Significance from a table" pages 381-383. This is merely a way to enable you to skip finding the p-value. I expect you to always find the p-value. So you don't need to read this section. You can do all the problems just by finding the p-value first.
   
   
10. What will you have to do on EVERY hypothesis testing problem? Answer: In addition to the obvious things, you must draw a picture of the sampling distribution of x-bar if Ho is true, and label it, and put the data value on it, and shade in the p-value. Also, you must write the conclusion at the end in terms of the actual real-life issue in the problem, not just "Reject Ho."
    
    
11. Write conclusions: "These data provide significant evidence, at the __% level, that ....... (fill in a statement of that says the Ha is true.)
    or
    "These data DO NOT provide significant evidence, at the __% level, that ....... (fill in a statement of that says the Ha is true.)
    
    
12. Exercise 14.57. This is a really useful idea. It explains how you can do a two-sided hypothesis test from a confidence interval. That will be really convenient in later chapters.

Chapter 15.

Read this chapter carefully. Omit the last section about "Planning studies: the power of a statistical test." Ask questions as needed about the homework.

Exercise 15.2. a. 1.92 +- 0.1209, which is 1.80 to 2.04 motorists. b. The Central Limit Theorem tells us that the sampling distribution of x-bar can be considered normal if the sample size is large enough. (Over n=30 or so.) This sample size is large enough. c. Very large non-response. The rate of non-response is (45,956-5029)/45,956 = 89.1%. That means we can't trust that the sample we got is representative of the population.

Exercise 15.4. The margins of error are
90%: 0.4824
95%: 0.5748
99%: 0.7555
So, as the confidence level increases, the margin of error increases, and the length of the interval increases.

Exercise 15.6.
a. Not included. This is a flaw in the sampling design.
b. Not included. This error arises from the sampling process.
c. Included: This random error is the only error addressed by confidence interval methods.

Exercise 15.8

Exercise 15.10.

a. In a sample of size 500, we expect, by chance, to see about 5 people who have high enough scores to have a p-value of 0.01 or less, because 500*0.01 = 5, even if the null hypothesis, of not having ESP, is true for all 500 people. So the four people we actually saw may have ESP, or they may be some of the "lucky" ones whom we would expect to see just by chance even if no one has ESP.
b. The researcher should do a similar experiment again on just those four people to see whether they show up as having ESP again. If their original results came from chance alone, we would not expect them to show up as having ESP again. So if they show up both times, then we consider that strong evidence that they have ESP.

Exercise 15.12.

n = (z * sigma / margin)^2 = (1.645 * 60 / 10)^2 = 97.42, so we need a sample of size 98.