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Confidence Intervals for Means and Proportions for AP Statistics

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Practice problems for these concepts can be found at:

In this lesson, we get more specific by actually constructing confidence intervals for each of the parameters under consideration. The chart below lists each parameter for which we will construct confidence intervals, the conditions under which we are justified in constructing the interval, and the formula for actually constructing the interval.

Special note concerning the degrees of freedom for the sampling distribution of the difference of two means: In most situations, a conservative, and usually acceptable, approach for determining the required number of degrees of freedom is to let df = min { n1 – 1, n2 – 1 }. This is "conservative" in the sense that it will give a smaller number of degrees of freedom than other methods, which translates to a larger margin of error. If you choose not to use the conservative approach, there are two cases of interest: (1) the population variances are assumed to be equal; (2) the population variances are not assumed to be equal (the usual case).

1. If we can justify the assumption that the population variances are equal, we can "pool" our estimates of the population standard deviation. In practice, this is rarely done since the statistical test for equal variances is unreliable. However, if we can make that assumption, then df = n1 + n2 – 2, and the standard error becomes . You will never be required to use this method (since it is very difficult to justify the assumption that the population variances are equal), although you should know when it is permitted.
2. The confidence interval can be constructed by calculator or computer (that's the "computed by software" notation in the chart). In this case, the degrees of freedom will be computed using the following expression:

You probably don't want to do this computation by hand, but you could! Note that this technique usually results in a noninteger number of degrees of freedom.

In practice, since most people will be constructing a two-sample confidence interval using a calculator, the second method above (referred to as the "computed by software" method in the box above) is acceptable. Just be sure to report the degrees of freedom as given by the calculator so that a reader knows that you used a calculator.

example: An airline is interested in determining the average number of unoccupied seats for all of its flights. It selects an SRS of 81 flights and determines that the average number of unoccupied seats for the sample is 12.5 seats with a sample standard deviation of 3.9 seats. Construct a 95% confidence interval for the true number of unoccupied seats for all flights.

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