Lesson 6 Homework Practice Compare Populations Answersl

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How to Compare Two Populations Using Sample Statistics

In this article, we will learn how to compare two populations using sample statistics, such as mean, standard deviation, and proportion. We will also learn how to make informal inferences about the difference between the populations based on the sample data. This is a useful skill for analyzing and interpreting data from surveys, experiments, or other sources.

Let's start with an example. Suppose we want to compare the average height of students in two different classes: Class A and Class B. We randomly select 20 students from each class and measure their heights. Here are the results:

ClassSample Mean (cm)Sample Standard Deviation (cm)

A165.38.7

B162.49.2

Calculate the difference between the sample means: 165.3 - 162.4 = 2.9 cm.

Estimate the standard error of the difference between the sample means: sqrt((8.7^2/20) + (9.2^2/20)) = 2.4 cm.

Use the standard error to construct a 95% confidence interval for the difference between the population means: 2.9 +/- 2 * 2.4 = (-1.9, 7.7) cm.

Interpret the confidence interval: We are 95% confident that the true difference between the average height of students in Class A and Class B is between -1.9 cm and 7.7 cm.

Make an informal inference: Since the confidence interval contains zero, we cannot conclude that there is a significant difference between the average height of students in Class A and Class B.

We can use a similar approach to compare two populations using sample proportions, such as the percentage of students who like chocolate ice cream in each class. The only difference is that we need to use a different formula for the standard error of the difference between the sample proportions:

SE(p1 - p2) = sqrt((p1 * (1 - p1) / n1) + (p2 * (1 - p2) / n2))

where p1 and p2 are the sample proportions, and n1 and n2 are the sample sizes.

For example, suppose we randomly select 20 students from each class and ask them if they like chocolate ice cream. Here are the results:

ClassSample Proportion (%)

A60

B50

How can we use these sample proportions to compare the two populations of students in Class A and Class B Here are some steps to follow:

Calculate the difference between the sample proportions: 0.6 - 0.5 = 0.1.

Estimate the standard error of the difference between the sample proportions: sqrt((0.6 * 0.4 / 20) + (0.5 * 0.5 / 20)) = 0.11.

Use the standard error to construct a 95% confidence interval for the difference between the population proportions: 0.1 +/- 2 * 0.11 = (-0.12, 0.32).

Interpret the confidence interval: We are 95% confident that the true difference between the percentage of students who like chocolate ice cream in Class A and Class B is between -12% and 32%.

Make an informal inference: Since the confidence interval contains zero, we cannot conclude that there is a significant difference between the percentage of students who like chocolate ice cream in a474f39169