AP Stats Unit 8 Chapter Notes, Lecture notes of Mathematics

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Chapter 8

Estimating Proportions

with Confidence

Section 8.

Estimating a Population

Proportion

By the end of this section, you should be able to:

LEARNING TARGETS

Estimating a Population Proportion

✔ STATE and CHECK the Random, 10%, and Large Counts

conditions for constructing a confidence interval for a population

proportion.

✔ DETERMINE the critical value for calculating a C% confidence

interval for a population proportion using a table or technology.

✔ CONSTRUCT and INTERPRET a confidence interval for a

population proportion.

✔ DETERMINE the sample size required to obtain a C% confidence

interval for a population proportion with a specified margin of

error.

There are three conditions that must be met for this formula to be valid—one for each of the three components in the formula.

There are three conditions that must be met for this formula to be valid—one for each of the three components in the formula.

There are three conditions that must be met for this formula to be valid—one for each of the three components in the formula.

When the standard deviation of a statistic is estimated from data, the result is called the standard error of the statistic.

When the standard deviation of a statistic is estimated from data, the result is called the standard error of the statistic.

In AP Statistics, confidence intervals come in the form point estimate ± margin of error

An equivalent form is statistic ± (critical value)·(standard error of statistic)

Conditions for Constructing a Confidence Interval

about a Proportion

Problem : Mr. Buckley’s class wants to construct a confidence interval for p = the true proportion of red beads in the container, which includes 3000 beads. Recall that the class’s sample of 251 beads had 107 red beads and 144 other beads. Check if the conditions for constructing a confidence interval for p are met.

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• Random: The class took a random sample of 251 beads from the container. ✓

Problem : Mr. Buckley’s class wants to construct a confidence interval for p = the true proportion of red beads in the container, which includes 3000 beads. Recall that the class’s sample of 251 beads had 107 red beads and 144 other beads. Check if the conditions for constructing a confidence interval for p are met.

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• Random: The class took a random sample of 251 beads from the container. ✓ º 10%: 251 beads is less than 10% of 3000. ✓

How do we get the critical value z* for our confidence interval?

How do we get the critical value z* for our confidence interval?

Finding the critical value z* for a 95% confidence interval starts by labeling the middle 95% under a standard Normal curve and calculating the area in each tail.

How do we get the critical value z* for our confidence interval?

Finding the critical value z* for a 95% confidence interval starts by labeling the middle 95% under a standard Normal curve and calculating the area in each tail.

Using Table A: Search the body of Table A to find the point – z * with area 0.025 to its left. The entry z =

• 1.96 is what we are looking for, so z * = 1.96.

Using technology: The command invNorm(area:0.025, mean:0, SD:1) gives z = –1.960, so z * = 1.960.

Problem : According to a 2016 Pew Research Center report, 73% of American adults have read a book in the previous 12 months. This estimate was based on a random sample of 1520 American adults. Assume the conditions for inference are met. (a) Determine the critical value z* for a 90% confidence interval for a proportion. (b) Construct a 90% confidence interval for the proportion of all American adults who have read a book in the previous 12 months. (c) Interpret the interval from part (b).

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