Constructing a Confidence Interval

Why settle for 95% confidence?

Conditions for Constructing a Confidence Interval

Random: The data come from a well-designed random sample or randomized experiment.

Normal: The sampling distribution of the statistic is approximately Normal

Independent: Individual observations are independent and the 10% condition is met.

The m&ms

Now for the challenge: you have about 10 minutes to find a 90% confidence interval for the proportion of red m&ms in the bag!

When Mr. Vignolini’s class did the m&ms Activity, they got 107 red m&ms and 144 others. Their point estimate for the unknown proportion p of red m&ms in the population is

426.0251

107ˆ p

We are 90% confident that the interval from 0.375 to 0.477 captures the actual proportion of red m&ms in Mr. Vignolini’s container.

n

pp )ˆ1(ˆ Standard error:

n

ppp

)1(ˆ

Mr. Vignolini claims that exactly half of the m&ms in the container are red. Use your result to comment on this claim.

The confidence interval in part (a) gives a range of plausible values for the population proportion of red m&ms. Since 0.5 is not contained in the interval, we have reason to doubt Mr. Vignolini’s claim.

n

ppzp

)ˆ1(ˆˆ *

Statistics (critical value).(standard deviation of statistics) Our confidence interval has the form:

To find a level C confidence interval, we need to catch the central area C under the standard Normal curve.

Find the critical value (z*) and an 80% confidence interval for the proportion of red m&ms in the bag!

z* = 1.28

n

ppzp

)ˆ1(ˆˆ *

One-Sample z Interval for a Population Proportion

the critical value for the standard Normal curve

Use this interval only when the numbers of successes and failures in the sample are both at least 10 and the population is at least 10 times as large as the sample

Alcohol abuse has been described by college presidents as the number one problem on campus, and it is an important cause of death in young adults. How common is it? A survey of 10,904 randomly selected U.S. college students collected information on drinking behavior and alcohol-related problems. The researchers defined “frequent binge drinking” as having five or more drinks in a row three or more times in the past two weeks. According to this definition, 2486 students were classified as frequent binge drinkers.

Binge drinking

1. Identify the population and the parameter of interest

2. Check conditions for constructing a confidence interval for the parameter

3. Find the critical value for a 99% confidence interval. Show your method. Then calculate the interval.

4. Interpret the interval in context

We are 99% confident that the interval from 21.8% to 23.8% captures the true proportion of U.S. college students who would be classified as binge drinkers

Putting It All Together: The Four-Step Process

Confidence Intervals: A Four-Step Process

State: What parameter do you want to estimate, and at what confidence level?

Plan: Identify the appropriate inference method. Check conditions

Do: If the conditions are met, perform calculations

Conclude: Interpret your interval in the context of the problem

AP EXAM TIP If a free-response question asks you to construct and interpret a confidence interval, you are expected to do the entire four step process. That includes clearly defining the parameter and checking conditions.

AP EXAM TIP You may use your calculator to compute a confidence interval on the AP exam. But there’s a risk involved. If you just give the calculator answer with no work, you’ll get either full credit for the “Do” step (if the interval is correct) or no credit (if it’s wrong). We recommend showing the calculation with the appropriate formula and then checking with your calculator. If you opt for the calculator-only method, be sure to name the procedure (e.g., one-proportion z interval) and to give the interval (e.g., 0.514 to 0.606).

Choosing the sample size (n)

for Desired Margin of Error when estimating μ

n

ppzME

)ˆ1(ˆ*

Given: Maximum margin of error ME

andConfidence interval C

Solve the inequality:

MEn

ppz

)ˆ1(ˆ* 5.0ˆ p

Suppose that you wanted to estimate the p = the true proportion of students at your school that have a tattoo with 95% confidence and a margin of error of no more than 0.10.

Problem: Determine how many students should be surveyed to estimate p within 0.10 with 95% confidence.

Tattoos

z* = 1.96 0.5 1 0.5

1.96 0.10n

2

1.960.5 1 0.5

0.10n

04.96n

So, we need to survey at least 97 students

Customer SatisfactionA company has received complaints about its customer service. The managers intend to hire a consultant to carry out a survey of customers. Before contacting the consultant, the company president wants some idea of the sample size that she will be required to pay for. One critical question is the degree of satisfaction with the company’s customer service, measured on a five-point scale. The president wants to estimate the proportion p of customers who are satisfied (that is, who choose either “satisfied” or “very satisfied,” the two highest levels on the five-point scale). She decides that she wants the estimate to be within 3% (0.03) at a 95% confidence level. How large a sample is needed?

we round up to 1068 respondents to ensure that the margin of error is no more than 3%.

Exercises on page 496-497 # 27,31-39 odd, 42, 47-52 all

Read Section 8.3