Unit 5 – Chapters 10 and 12

1. Unit 5 – Chapters 10 and 12 • What happens if we don’t know the values of population parameters like and ? • Can we estimate their values somehow? • Can we find a range in which their values might lie?

2. Page 2 – Chapters 10&12 vocabulary (1) Statistical Inference Typically, we don’t know the values of parameters – usually and . So we will use statistics – results from samples – to estimate the values of these parameters. (2) Statistics are variables, of course, so each value of and p-hat that we get is a single point on the number line that it could fall on. > These values, then, are called point estimates.

3. Page 3 – Chapter 10 vocabulary (3) Since our point estimates are variable, there will be some imprecision associated with them. • Thus, we will use our statistics to create an interval in which we think that the value of the parameter will be. • The formula for a Confidence Interval for : This part is called the Margin of Error of our Confidence Interval.

4. Page 4 – Chapter 10 vocabulary (4) What happens to the Margin of Error of our Confidence Interval if …… • “n” is increased? • The confidence level is increased? This part is called the Margin of Error of our Confidence Interval.

5. Page 5 – Chapter 10 vocabulary (5) What is the interpretation of a Confidence Interval? • An example: “I am 95% confident that the mean height of all American young women is between 64.8 and 65.6 inches.” • NOT THIS ONE: “There is a 95% chance that the mean height of all American young women is between 64.8 and 65.6 inches.” • NOR THIS ONE: “95% of all American young women have heights between 64.8 and 65.6 inches.” • NOTE: A Confidence Interval interpretation must always be in the context of the problem. Mr. T: “I would say that memorizing this phrase would be a good idea….”

6. Page 6– Chapter 10 vocabulary (5) (continued) • These Confidence Intervals will be different every time we calculate one, of course, since they are based around a statistic – a variable value. • So, how many of them will actually capture the true parameter value? • The interpretation of the Confidence Level: “95% of all intervals created in this manner will contain the true mean height of all American young women.”

7. Page 7 – Chapter 10 vocabulary • (6) If we have an idea of the true value of a parameter and want to test the validity of our belief about this value, then we will perform a HYPOTHESIS TEST. • The evidence from the sample(s) will either support our belief or make us question it. (7) All Hypothesis Tests are set up the same way, no matter what chapter, distribution, or parameter symbol we are using:

8. Page 8 – Chapter 10 vocabulary (7) An example of how Hypothesis Tests should begin: Suppose we think that the mean height of all American young women is 65.5 inches, but some scientists suspect that this value has actually increased in the last generation or so (due to better nutrition, decreasing smoking rates, etc.). The null hypothesis, , is our current belief. It always has an = in it and always is about a parameter – no exceptions to this!

9. Page 9 – Chapter 10 vocabulary • The alternative hypothesis, , is our suspicion about the true value of the parameter in question. It could be a >, like this: We suspect that the mean height has increased. It could be a <, like this: We suspect that the mean height has decreased. It could be the case that we suspect that the mean height has changed, but we’re not sure which way:

10. Page 10 – Chapter 10 vocabulary (8) In order to see whether our current belief, , about the value of a parameter is support by the sample evidence or not, we will rely on something called the p-value. The p-value is the probability that, if the null hypothesis is true, we would have gotten sample evidence as extreme as we did. Small p-values mean one of two things: • The null hypothesis is true, but, just because of bad luck, the sample results were improbable, or • This improbability is too much for us to believe, and we will instead reject the null hypothesis.

11. Page 11 – Chapter 10 vocabulary (9) How small does the p-value have to be in order for us to lose faith in the null hypothesis? This level is generally referred to as (“alpha”), and the default value for is 0.05. If the p-value < alpha, then we will “reject” . If the p-value > alpha, then we will “fail to reject” . Mr. T says, “Please always use one of these two phrases when making a conclusion from a Hypothesis Test.”

12. Page 12 – Chapter 10 vocabulary (10) How can we remember all of the steps involved in performing a Hypothesis Test? HCCC H = write your two Hypotheses C = check the Conditions C = perform the Calculations C = write a Conclusion

13. Page 13 – Chapter 10 vocabulary • (11) Section 10.4 – Inference Errors • What if, when we make a conclusion from a Hypothesis Test, we make an error? • For example, what if is actually true, but we reject it? • This is called a Type I Error. • What is the probability that we make a Type I Error?

14. Page 14 – Chapter 10 vocabulary • For example, we are doing our usual: • Our sample happens to have several tall-than-usual young women in it, which makes the p-value small, and we reject the null hypothesis; • That is, based on the sample data, we conclude that the popn mean height is greater than 65.5”, when in fact the popn mean is reallyequal to 65.5”.

15. Page 15 – Chapter 10 vocabulary • There are four situations that could occur: The null hypothesis is actually true – but we reject it. This is a Type I error, and it is bad. The null hypothesis is actually false, and we reject it based on our sample data. Yay! We made the right decision! The null hypothesis is actually false, but we fail to reject it. This is a Type II error – it is bad. (Note: calculating its probability is hard) The null hypothesis is actually true, and we fail to reject it. Yay! We made the right decision!

16. Page 16 – Chapters 10-12 vocabulary A review of this thing called “INFERENCE” • Used to try and gain some knowledge about an unknown population parameter (usually µ or p) • If we don’t know the value of a parameter, we might try to… > calculate a range of believable values for it (this is a Confidence Interval) or > test it against a specific value that we think may or may not be true (this is a Hypothesis Test).

17. Page 17 – Chapter 12 vocabulary (12) What if it’s a population proportion that we don’t know? • For example, you are working for the re-election campaign of Senator Snodwhistle, who naturally wants to know if he will win. Specifically, he wants to know if p – the true proportion of all voters who will pick him – is equal to 0.50 or greater than 0.50. • What would the hypotheses be? How will we decide? We will take a sample, calculate its p-hat, and compare this to 0.5 How are these p-hats distributed?

18. SECTION 9.2 VOCABULARY – pg 6 These p-hats – which vary, since they are statistics – can be graphed, of course, and, IF certain conditions are met, then we know what sort of shape they will make. (7) IF all of these four conditions are met: * the sample was an SRS, * the size of the population is >10n, * the value of np is >10, and * the value of n(1-p) is >10, THEN the sampling distribution of p-hat is approximately Normal. (This is similar to the Central Limit Theorem, but not called that.)

19. Page 19 – Chapter 12 vocabulary The good news is that – regardless of what chapter we are in or what parameter we are investigating – the four steps for a Hypothesis Test remain the same! HCCC H = write your two Hypotheses C = check the Conditions C = perform the Calculations C = write a Conclusion

20. Page 20– Chapter 12 vocab – LAST ONE! (3) Since our point estimates are variable, there will be some imprecision associated with them. • Thus, we will use our statistics to create an interval in which we think that the value of the parameter will be. • The formula for a Confidence Interval for : This part is called the Standard Error of phat This part is still called the Margin of Error of our Confidence Interval