AP Statistics – 9.2 Name: ____________________ Confidence Intervals and Power of Significance Tests

1. Use a confidence interval to draw a conclusion for a two-sided test about a population parameter. The result of a significance test is basically a decision to reject H0 or fail to reject H0. When we reject H0, we’re left wondering what the actual proportion p might be. A confidence interval might shed some light on this issue. Example Problem: Edward claims that half of the students at his school have never vaped. He found that 90 of an SRS of 150 students said that they had never vaped. The number of successes and the number of failures in the sample are 90 and 60, respectively, so we can proceed with calculations. Create a 95% confidence interval and interpret its meaning. Our 95% Confidence interval is:

�̂� ± 𝒛∗√�̂�(𝟏 − �̂�)

𝒏

𝟎. 𝟔𝟎 ± 𝟏. 𝟗𝟔√𝟎. 𝟔𝟎(𝟎. 𝟒𝟎)

𝟏𝟓𝟎

(𝟎. 𝟓𝟐𝟐, 𝟎. 𝟔𝟕𝟖)

Interpretation: We are 95% confident that the interval from 0.522 to 0.678 captures the true proportion of students at Edward’s high school who would say they have never used a vaping device. Analysis: Is there convincing evidence at the 5% significance level that the Edward’s claim is incorrect? Justify your answer. Why Confidence Intervals Give More Information There is a link between confidence intervals and two-sided tests. The 95% confidence interval gives an approximate range of p0’s that would not be rejected by a two-sided test at the α = 0.05 significance level.

9 A two-sided test at significance level α (say, α = 0.05) and a 100(1 –α) % confidence interval (a 95% confidence interval if α = 0.05) give similar information about the population parameter.

Key

HoiP 50Ha Pt 50

becausethe Ho doesnot fallwithin the 95 confidence interval we haveconvincingevidence that Edwardsclaim is incorrectWerejectHo

AP Statistics – 9.2 Name: ____________________ Confidence Intervals and Power of Significance Tests

2. Describe the relationship among the probability of Type I error (significance level), the probability of Type II error, and the power of a test.

Type II Error and the Power of a Test

x A significance test makes a Type II error when it fails to reject a null hypothesis H0 that really is false.

x There are many values of the parameter that make the alternative hypothesis Ha true, so we concentrate on one value.

Power = the probability of correctly rejecting the null Potato Chip Blemishes Revisited A potato chip producer and its main suppliers agree that each shipment of potatoes must meet certain quality standards. If the producer finds convincing evidence that more than 8% of the potatoes in the shipment have blemishes, the truck will be sent away to get another load from the supplier. A supervisor selects a random sample of 500 potatoes from the truck. An inspection reveals that 47 of the potatoes have blemishes. The potato-chip producer wonders whether the significance test of

H0 : p = 0.08 Ha : p > 0.08

based on a random sample of 500 potatoes has enough power to detect a shipment with, say, 11% blemished potatoes. In this case, a particular Type II error is to fail to reject H0 : p = 0.08 when p = 0.11. Type II Error and the Power of a Test

α = The Probability of making Type I Error β = The Probability of making Type II Error How to increase the power of a Test:

1. Increase the Sample Size 2. Increase the Significance Level 3. Increase the Effect Size (difference between the null

and alternative parameter values)

The power of a test against a specific alternative is the probability that the test will reject H0 at a chosen

significance level α when the specified alternative value of the parameter is true.

The power of a test against any alternative is 1 minus the probability of a Type II error for that alternative; that is, power = 1 − β.

AP Statistics – 9.2 Name: ____________________ Confidence Intervals and Power of Significance Tests

Practice Problem: 1) Which more serious for the potato-chip producer in this setting: a Type I error or a Type II error?

Based on your answer, would you choose a level of α = 0.01, 0.05, or 0.10? 2) Tell if each of the following would increase or decrease the power of the test. Justify your answers.

a) Change the significance level to α = 0.10

b) Take a random sample of 250 potatoes instead of 500 potatoes.

c) Insist on being able to detect that p = 0.10 instead of p = 0.11