Warm-upDay of 8.1 and 8.2 Quiz and Types of Errors Notes

Answers to Inference and Proportions Worksheet1.The estimate is 0.86, SE = 0.0347 so the 95% CI is0.7920 to 0.9280.2.a) H0: p = .384 HA: p > .384 where p is the proportion of all

free throws that Sue makes this season.b) The test statistic is Z = 3.13.c) The P-value is 0.0009. Reject the null hypothesis for a =

0.05; also at a = 0.01.d) .4991 to .7509. There is strong evidence that Sue has

improved.e) We must assume that the shots are equivalent to a random

sample from all shots. Also, the sample size must be sufficiently large (which it is).

Answers continued…3.The estimate is .173 with SE = .0437, so the 95% CI is

0.0877 to 0.2590.Answer to A.P. Questiona.

Answer to a.of A.P. Question continued…Confidence Interval

Interpretation of interval

Interpretation of confidence level

Answers to parts b. and c. b.

c.

8.2 Type of ErrorsThe logic of guilty or not guilty works when thinking aboutwhether to reject or fail to reject the null hypothesis.

When understanding errors in hypothesis testing, itis best to think of medical tests to determine if adisease is present.

Other ways to remember.

Normally you start with the idea that the null hypothesis is true, andthat the person is healthy. So your first mistake would be to reject thenull hypothesis or to declare the person diseased. This would be yourType I error.When do Type I errors occur?They occur when you have the bad luck of drawing an unusualsample. Also when you choose a significance level (α) this also createsthe probability of making a Type I error.

Power and Type II Errors βA test’s ability to detect a false hypothesis is its power.When the null hypothesis is actually false, we hope ourtests is strong enough to reject it. So β is the probabilitythat the test fails to rejected a false null hypotheses.Power = 1- β . Whenever a study fails to reject its null hypothesis, the test’s power comes into question.

Reducing both Type I and Type II errorsReducing sample size is the best way to reduce a

Type II error.

The greater likelihood of Type I error, means thetest has more power.

Discussion QuestionsPg 506 of textbookD36. Explain why an increase in sample size increases

the power of a test, all else remaining unchanged.D37. What happens to the power of a test as the

population proportion, p, moves farther away from the hypothesized value, p0, all else remaining unchanged?

D38. Recall that the power of a test is the probability of rejecting the null hypothesis. Can a statistical test of the type discussed in this chapter ever have a power of 1? Can a statistical test of the type discussed in this chapter ever have a power of 0? If so, would either be desirable from a practical point of view

Vocabulary you should have for 8.1

8.1 level of confidence(capture rate), margin of error8.2 significance test for a proportion, statistical

significance, null hypothesis, alternate hypothesis, test statistic, p-value, critical values, level of significance, type 1 error, type 2 error, power of a test, one-sided, two-sided test

17 terms for 8.1 and 8.2

Directions for Quiz

• Read the problems carefully.• Follow the steps reviewed in class for #2.• For #2, set up the alternate hypothesis as not

equal to the proportion pH A