Inference on Proportions

What are the steps for performing a confidence interval?

1. Assumptions

2. Calculations

3. Conclusion

1. Assumptions:

• SRS of context

• Approximate Normal distribution because

np > 10 & n(1-p) > 10

• Population is at least 10n(tests independence)

2. Formula for Confidence interval:

statistic of SD valuecritical statisticCI

p̂

npp 1*z

Normal curve

Note: For confidence intervals, we DO NOTDO NOT know p – so we MUST substitute p-hatp-hat for pin both the SD & when checking assumptions.

3. Conclusion Statement

• We are X % confident that the true proportion of whatever is happening in context of the problem is between A and B.

A January 2015 poll that Ms. Nichols totally made up a random sample of 2015 high school AP students, 814 said that Calculus is more fun than Statistics. Find a 90% confidence interval for the true proportion of AP students who think that Calculus is more fun than Statistics.

Assumptions:

SRS of AP students YES

Approx. Normal:

2015 (814/2015) > 10 YES

2015 (1-814/2015) > 10 YES

Independence:

pop. >10(2015) YES

Calculations:

Conclusion:

We are 90% confident that the true proportion of AP students who think Calculus is more fun that Statistics is between .3860 and .4220.

4220,.3860.2015

)2015814

1(2015814

6448.12015

8141*ˆ

n

ppzP

A May 2000 Gallup Poll found that 38% of a random sample of 1012 adults said that they believe in ghosts. Find a 95% confidence interval for the true proportion of adults who believe in ghost.

Assumptions:

•Have an SRS of adults

•np =1012(.38) = 384.56 & n(1-p) = 1012(.62) = 627.44 Since both are greater than 10, the distribution can be approximated by a normal curve

•Population of adults is at least 10,120.

41,.35.1012

)62(.38.96.138.

1*ˆ

npp

zP

We are 95% confident that the true proportion of adults who believe in ghosts is between 35% and 41%.

Step 1: check assumptions!

Step 2: make calculations

Step 3: conclusion in context

Another Gallop Poll is taken in order to measure the proportion of adults who approve of attempts to clone humans. What sample size is necessary to be within + 0.04 of the true proportion of adults who approve of attempts to clone humans with a 95% Confidence Interval?

To find sample size:

However, since we have not yet taken a sample, we do not know a p-hat (or p) to use!

npp

zm1

*

What p-hat (p) do you use when trying to find the sample size for a given margin of error?

.1(.9) = .09

.2(.8) = .16

.3(.7) = .21

.4(.6) = .24

.5(.5) = .25

By using .5 for p-hat, we are using the worst-case scenario and using the largest SD in our calculations.

Another Gallop Poll is taken in order to measure the proportion of adults who approve of attempts to clone humans. What sample size is necessary to be within + 0.04 of the true proportion of adults who approve of attempts to clone humans with a 95% Confidence Interval?

60125.600

25.96.104.

5.5.96.104.

5.5.96.104.

1*

2

n

n

n

n

npp

zm

Use p-hat = .5

Divide by 1.96

Square both sides

Round up on sample size

Stop & do homework!

Steps for a Hypothesis Test

1. Check Assumptions (same as Conf. Int.)

2. State Hypotheses

3. Calculate the test statistic (Z)

4. Calculate the p-value

5. State Conclusion

Hypotheses for proportions:

H0: p = value

Ha: p > value

where p is the true proportion of context

Use >, <, or ≠

Formula for hypothesis test:

statistic of SD

parameter - statisticstatisticTest

z npp

pp

1

ˆ

Calculating the P-Value

• The p-value is the probability that you get your result by chance alone assuming the null hypothesis is true

• Use normalcdf

• Let mean = 0, SD = 1

• For less than: z = UB

• For greater than z = LB

Conclusion StatementIf the p-value < α…

Since the p-value is less than α we reject the null hypothesis. There is sufficient evidence to suggest that the true proportion of what we are calculating the proportion of in the problem is greater/less/different than original p in problem.

If the p-value is > α…

Since the p-value is greater than α we fail to reject the null hypothesis. There is not sufficient evidence to suggest that the true proportion of what we are calculating the proportion of in the problem is greater/less/different than original p in problem.

Note: Unless otherwise stated in the problem assume that α is 0.05

α is also called the significance level

A company is willing to renew its advertising contract with a local radio station only if the station can prove that more than 20% of the residents of the city have heard the ad and recognize the company’s product. The radio station conducts a random sample of 400 people and finds that 90 have heard the ad and recognize the product. Is this sufficient evidence for the company to renew its contract?

Assumptions:

•Have an SRS of people

•np = 400(.2) = 80 & n(1-p) = 400(.8) = 320 - Since both are greater than 10, this distribution is approximately normal.

•Population of people is at least 4000.

H0: p = .2 where p is the true proportion of people who

Ha: p > .2 heard the ad

05.α1056.25.1

400)8(.2.

2.225.

valuepz

Since the p-value >, I fail to reject the null hypothesis. There is not sufficient evidence to suggest that the true proportion of people who heard the ad is greater than .2.

Use the parameter in the null hypothesis to check assumptions!

Use the parameter in the null hypothesis to calculate standard

deviation!

A new reality TV show, “I Want to Marry a Statistician,” has been showing on Monday evenings, and ratings showed that it has been watched by 55% of the viewing audience each week. The producers are moving the show to Wednesday night. They are interested in whether or not the move had affected the ratings. After the show was moved, a random sample of 500 people who watch TV on Wednesday night are surveyed and asked what show they are watching. 255 respond with “I Want to Marry a Statistician.” Were the ratings affected by the move at the .1 level of significance?

Assumptions:

SRS of viewers YES

500(.55) > 10 and 500(.45) > 10 YES

Pop. of TV watchers > 5000 YES

H0: p = .55 Where p is the true proportion of people Ha: p ≠ .2 watching the show

Since the p-value is less than α I reject the null hypothesis. There is sufficient evidence to suggest that the proportion of people watching the show on Wednesday is different from what it was on Monday.

1.0722.2*0361.7979.1

500)45(.55.

55.500255

valuepz

For hypothesis testing when you DON’T need to do a complete

write up….• H0 and Ha

• The test statistic (Z)

• The p-value

• Reject or fail to reject H0

• Answer the question (yes or no)

Note: You must do a complete write-up on the AP test