10 1 Confidence Intervals by 52S4R6


									AP Statistics
Chapter 10-1
Lecture Notes

                                    INTRO TO CONFIDENCE INTERVALS
There are two main types of inference taught in AP Statistics, confidence intervals and significance test
(hypothesis test). The basic idea of a confidence interval (CI) is this. I have an estimate, either a sample mean
or a sample proportion, if my assumptions of independence (verified by c 1 and c3) and normality (verified by c2)
can be justified, then I can create an interval centered on my estimate where it is likely that the “true but
unknown” parameter is one of the values in the interval. The basic form a confidence interval is this:

                                             Estimate  Margin of Error
                                                 Statistic  z*statistic
 The value z * represents the number of standard deviations (of a normal distribution) that the interval extends
on both sides of the estimate, which corresponds to a certain probability. It is this value that gives us a
confidence level (C) in the method we used to construct the interval. The bigger the z * gets the larger the
CI gets, giving us a better probability of “capturing” the true parameter. The z * values can be found in table C
in the back of your book; the most commonly used values are below.

                                              z*                 C
                                            1.645              90%
                                            1.960              95%
                                            2.576              99%
                                            2.807             99.5%
                                            3.291             99.9%

Before doing an example let’s explore the margin of error further. The margin of error is made up of two
components, the confidence ( z * ), and the standard error of the statistics sample distribution. You should
know how each component can increase or decrease the length of the interval.

                CI for Sample Mean ( x )                                                         
                                                                      CI for Sample Proportion ( p )
                                                                                        pq
                       x  z*                                                   p  z*
                                n                                                        n

Example: The Gallup organization conducted a poll about Halloween practices and beliefs during October
21-24, 1999. A sample of 1005 adult Americans were asked whether someone in their family would give out
Halloween treats from the door of their home, and 69% answered yes.

         a)   Is 0.69 a parameter or a statistic?
         b)   Does the question asked by Gallup constitute a quantitative or a categorical variable?
         c)   Before calculating a confidence interval, we need to assume that our sample distribution of
              sample proportions is independent and normal. Verify these assumptions with the appropriate

         d)   Construct a 95% confidence interval for , the proportion of all adult Americans who planned to
              give out Halloween treats from their home in 1999.
         e)   In the same Gallup poll, a sample of 493 adults was asked whether they believe in witches, and
              22% said yes. Construct a 95% CI for this statistic

         f)   For the poll question about witches, suppose that the sample size had been 1005 instead of 493,
              and that 22% had answered yes. Would you expect a 95% confidence interval based on this
              larger sample to be wider or narrower than the one you found in (e)?

Example: Direct mail advertisers send solicitations to thousands of potential customers in the hope that some
will buy the company’s product. The response rate is usually quite low. Suppose a company wants to test the
response to a new flyer, and sends it to 1000 people randomly selected from their mailing list of over 200,000
people. They get 123 of the recipients

         a)   Verify the assumptions of independence and normality with the three necessary conditions.

         b)   Create a 90% confidence interval for the percentage of people the company contacts that buy

         c)   Explain what this interval means.

         d)   Explain what “90% confident” means.

         e)   The company must decide whether to now do a mass mailing. The mailing won’t be cost-
              effective unless it produces at least a 5% return. What does your confidence interval suggest?

Main Concepts: Confidence Intervals

        The confidence is in the method, not in the result.

        Confidence intervals are computed from random samples and therefore they are random. The
         parameter is not random.

        The parameter is fixed (but unknown), and the estimate of the parameter is random (but observable).
         If the estimate is likely to be within two standard errors of the parameter, then the parameter is likely
         to be within two standard errors of the estimate. This is the foundation on which confidence intervals
         are based.

        Confidence intervals can be used to check the reasonableness of claims about the parameter. If
         someone claims the parameter is equal to 62, and 62 is not within your confidence interval, than this
         claim is suspect. This type of thinking will be made more formal and precise later.

        There is a famous incorrect interpretation of a confidence interval: there is a 95% probability that
         the true mean caloric content lies between 260 and 310.

        Runner-up for most famous incorrect interpretation of a confidence interval: if we take repeated
         samples, 95% of them will have the sample mean between 260 and 310.

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