ARE YOU READY FOR THE QUIZ?

1. Yes, I’ve been working hard.2. Yes, I like this material on hypothesis test.3. No, I didn’t sleep much.4. No, some other reason.5. I guess we will find out.

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CHAPTER 22Comparing Two Proportions

CONFIDENCE INTERVALS FOR PROPORTION DIFFERENCES

1 1 2 21 2

1 2

ˆ ˆ ˆ ˆˆ ˆ

p q p qSE p p

n n

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When the conditions are met, we are ready to find the confidence interval for the difference of two proportions:

The confidence interval is

where

The critical value z* depends on the particular confidence level, C, that you specify.

1 2 1 2ˆ ˆ ˆ ˆp p z SE p p

HW 10 – PROBLEM 5

A study examined parental influence on teenage smoking.

A group of students who’d never smoked were asked about their parents attitude.

A year later they were asked if they had started smoking.

Parental attitude- Disapproved – 54 out of 286 smoked Lenient – 11 out of 38 smoked

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HW 10 – PROBLEM 5

Create a 95% confidence Interval

Interpret that interval

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Consider the 95% level: There’s a 95% chance that p is no more than 2 SEs

away from . So, if we reach out 2 SEs, we are 95% sure that p will

be in that interval. In other words, if we reach out 2 SEs in either direction of , we can be 95% confident that this interval contains the true proportion.

This is called a 95% confidence interval.

A CONFIDENCE INTERVAL

p̂

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p̂

A CONFIDENCE INTERVAL (CHANGING OUR INTERPRETATION)

Consider the 95% level: There’s a 95% chance that p1-p2 is no more than

2 SEs away from our observed difference. So, if we reach out 2 SEs, we are 95% sure that p1-p2 will be in that interval. In other words, if we reach out 2 SEs in either direction of our observed difference, we can be 95% confident that this interval contains the true proportion.

This is called a 95% confidence interval.

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WHAT IS THE 95% CI?1. The true difference lies in

the interval of more than 95% of all random samples

2. The true difference is probably in the CI

3. 95% of all random samples produce intervals that contain the true difference

4. The true difference is less than 5% from the confidence interval

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WE ARE 95% CONFIDENT… The proportion of teens with lenient parents who’ll

later smoke is 5% less to 25.2% more than for teens whose parents disapproved.

About 5% of teens whose parents disapproved will later smoke and 25.2% of teens with lenient parents will someday smoke

5% to 25.2% less teens whose parents disapproved than teens with lenient parents will later smoke

The proportion of teens whose parents disapproved who will later smoke is 5% less to 25.2% more than for teens with lenient parents

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WE ARE 95% CONFIDENT… The proportion of teens with lenient parents who’ll

later smoke is 5% less to 25.2% more than for teens whose parents disapproved.

About 5% of teens whose parents disapproved will later smoke and 25.2% of teens with lenient parents will someday smoke

5% to 25.2% less teens whose parents disapproved than teens with lenient parents will later smoke

The proportion of teens whose parents disapproved who will later smoke is 5% less to 25.2% more than for teens with lenient parents

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WE ARE 95% CONFIDENT… The proportion of teens with lenient parents who’ll

later smoke (p1) is 5% less to 25.2% more than for teens whose parents disapproved.

About 5% of teens whose parents disapproved will later smoke and 25.2% of teens with lenient parents will someday smoke

5% to 25.2% less teens whose parents disapproved than teens with lenient parents will later smoke

The proportion of teens whose parents disapproved who will later smoke is 5% less to 25.2% more than for teens with lenient parents

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WE ARE 95% CONFIDENT… The proportion of teens with lenient parents who’ll

later smoke (p1) is 5% less to 25.2% more than for teens whose parents disapproved.

About 5% of teens whose parents disapproved will later smoke and 25.2% of teens with lenient parents will someday smoke

5% to 25.2% less teens whose parents disapproved than teens with lenient parents will later smoke

The proportion of teens whose parents disapproved who will later smoke is 5% less to 25.2% more than for teens with lenient parents

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WE ARE 95% CONFIDENT… The proportion of teens with lenient parents who’ll

later smoke (p1) is 5% less to 25.2% more than for teens whose parents disapproved (p2).

About 5% of teens whose parents disapproved will later smoke and 25.2% of teens with lenient parents will someday smoke

5% to 25.2% less teens whose parents disapproved than teens with lenient parents will later smoke

The proportion of teens whose parents disapproved who will later smoke is 5% less to 25.2% more than for teens with lenient parents

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WE ARE 95% CONFIDENT…1. The proportion of teens with lenient parents who’ll

later smoke (p1) is 5% less to 25.2% more than for teens whose parents disapproved (p2).

2. About 5% of teens whose parents disapproved will later smoke and 25.2% of teens with lenient parents will someday smoke

3. 5% to 25.2% less teens whose parents disapproved than teens with lenient parents will later smoke

4. The proportion of teens whose parents disapproved who will later smoke (p2) is 5% less to 25.2% more than for teens with lenient parents (p1) Slide

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WE ARE 95% CONFIDENT…1. The proportion of teens with lenient parents who’ll

later smoke (p1) is 5% less to 25.2% more than for teens whose parents disapproved (p2).

2. About 5% of teens whose parents disapproved will later smoke and 25.2% of teens with lenient parents will someday smoke

3. 5% to 25.2% less teens whose parents disapproved than teens with lenient parents will later smoke

4. The proportion of teens whose parents disapproved who will later smoke (p2) is 5% less to 25.2% more than for teens with lenient parents (p1) Slide

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25%

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TWO-PROPORTION Z-TEST (CONT.)

1 21 2

ˆ ˆ ˆ ˆˆ ˆ pooled pooled pooled pooled

pooled

p q p qSE p p

n n

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We use the pooled value to estimate the standard error:

Now we find the test statistic:

When the conditions are met and the null hypothesis is true, this statistic follows the standard Normal model, so we can use that model to obtain a P-value.

1 2

1 2

ˆ ˆ 0

ˆ ˆpooled

p pzSE p p

HW 10 – PROBLEM 9

A study investigated whether regular mammograms resulted in fewer deaths from breast cancer.

Women would never had mammograms, 30,761, only 197 died of breast cancer.

Women who had mammograms, 30,360, only 162 died of breast cancer.

Do these results suggest mammograms reduce breast cancer deaths? (Test at significance level=0.01) Slide

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WHAT IS OUR HYPOTHESIS? WE WANT TO KNOW IF SCREENINGS IMPROVE (OR LOWER) THE DEATH RATE

1. Ho: p1 – p2 =0 Ha: p1 – p2>02. Ho: p1 – p2 =0 Ha: p1 – p2<03. Ho: p1 – p2 =0 Ha: p1 – p2≠0

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AT SIGNIFICANCE OF 0.01, WHAT IS YOUR TEST RESULT?

1. Reject Null. There is enough evidence to support the claim of a difference.

2. Accept Null. There is NOT enough evidence to support the claim of a difference.

3. Fail to Reject the Null. There is NOT enough evidence to support the claim of a difference.

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UPCOMING IN CLASS

Quiz #5 today.

Homework #10 due Sunday

Exam #2 is Wed. Nov 28th