Confidence IntervalsA confidence interval is an interval that a statistician

hopes will contain the true parameter value. A level C confidence interval means that C% of all intervals created by random samples on n will contain the parameter. A confidence interval looks like:

estimate margin of error or statistic (critical value)(sampling error)

Specifically, for a sample proportion a confidence interval is: )(ˆ * SEzp

Critical ValuesCritical values are the Z-scores that represent the

desired %. You can use the inverse normal (invnorm) command on your calculator.

% Z*

90% 1.64

95% 1.96

98%

99%

Invnorm((1 – C)/2) = critical value

More facts about confidence intervals?

• What does a 90% confidence interval mean?

90% confident means that 90% of all intervals done with a specific sample size will contain the true parameter.

• What is the margin of error?

The margin of error is the critical value times the SE. It is the amount that you are adding and subtracting to the statistic.

•What are the conditions/requirements to do confidence intervals?

The conditions or requirements that are required are the same conditions/requirements for sample proportions and sample means studied in the last chapter.

What happens to the margin of error when the confidence increases?

The margin of error would increase because the critical value would increase!

For example, z* = 1.64 for a 90% confidence interval while z* = 2.33 for a 98% confidence interval.

Ex 1: In a random sample of 140 teenagers in Placer County, 35 stated that they smoke regularly.

a) What is the parameter?

b) Are the requirements (assumptions) met to calculate a valid confidence interval?

c) Construct a 95% confidence interval for the true proportion of Placer county teenagers that smoke. Interpret your interval.

d) What is the margin of error for your confidence interval?

?p̂

In 1992 presidential election Bill Clinton ran against George H. W. Bush and Ross Perot. That June the Gallup organization asked registered voters if there was “Some chance they could vote for other candidates” besides their expressed first choice. At that time, 62% of registered voters said “yes,” there was some chance they might switch. In June 2004, Gallup/CNN/USA Today asked 909 registered voters the same question. Only 18% indicated that there was some chance they might switch. The resulting 95% confidence interval is 0.18 0.025 = 20.5%. Are these statements about the 2004 presidential election correct? Explain.

1. In the sample of 909 registered voters, somewhere between 15.5% and 20.5% of them said there is a chance they might switch votes.

2. We are 95% confident that 18% of all US registered voters had some chance of switching votes.

3. We are 95% confident that between 15.5% and 20.5% of all US registered voters had some chance of switching votes.

4. We know that between 15.5% and 20.5% of all US registered voters had some chance of switching votes.

5. 95% of all US registered voters had some chance of switching votes.

Example #3Cloning: A May 2002 Gallup Poll found that only 8% of a random sample of 1012 adults approved of attempts to clone a human.

a) Find the margin of error for this poll if we want 95% confidence in out estimate of the percent of American adults who approve of cloning humans.

b) Explain what that margin of error means.

c) If we only need 90% confident, will the margin of error be larger or smaller? Explain.

d) Find that margin of error.

e) In general, if all other aspects of the situation remain the same, would smaller samples produce smaller or larger margins of error?