A downloadable version of this activity is available in the following format:

### Purpose

The purpose of this activity is to investigate the relationship between the standard error and the population standard deviation.

### Outcomes

This activity will teach students how to

- apply characteristics of normal distributions
- demonstrate an understanding of the role of the central limit theorem in the development of confidence intervals
- demonstrate an understanding of how the size of a sample affects the variation in sample results
- graph and interpret sample distributions of the sample mean and sample distributions of the sample proportion

### Format

This activity will take the form of a teacher-led full-class discussion. Students are encouraged to share information. To facilitate discussion, divide students into small groups.

### Introduction

It is important that students realize that we are usually unable to collect information about a total population. The goal of sampling is to draw reasonable conclusions about a population by obtaining information from a relatively small part (a sample) of that population. In order to do that, we need to know what formulas to use and what degree of confidence we can claim in the resulting information.

The previous lesson on investigating sampling was a first step toward developing the concept of confidence intervals.

### Classroom instruction

**1. Normal distribution**

Students should recall the percentages associated with normal distributions:

Roughly: 68% of the data fall within one standard deviation of the mean

95% of the data fall within two standard deviations of the mean

99% of the data fall within three standard deviations of the mean

The concept of ‘within’ may require some discussion (between one standard deviation above the mean and one standard deviation below the mean). A diagram can help in visualizing this.

Using 164.5 as the mean and 2.28 as the standard deviation, students can examine the sample means in Appendix A. ‘Within one standard deviation of the mean’ gives the following calculation:

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Students can examine the sample means given in Appendix A to determine how many of them fall within the interval from 162.2 to 166.8. (Answer: 67 sample means fall in that interval, with 97 sample means falling within the interval 159.9 to 169.1.) The results are close to what one would expect from a normal distribution. From the histogram, we recognized that the distribution was not perfectly normal.

**2. The central limit theorem**

Have the students discuss their answers in small groups or as a class.

**Answers to the exercise:**

Sample 2: The interval 157.6 to 165.0 contains the population mean.

Sample 3: The interval 164.6 to 172.0 contains the population mean.

Sample 4: The interval 165.5 to 172.9 *does not* contain the population mean.

In working through this exercise, students recognize that when an interval is calculated as described, it will ‘very likely’ contain the population mean.

**3. Project time**

Ensure that students recognize the characteristics of the population from which they have chosen a sample. For example, is the population all Grade 11 students in the school, or is it all students taking Grade 11 Mathematics in the school? Is it all students in the school district, in the province, or in the whole of Canada? It may also be useful to discuss alternative ways of describing the confidence level. Here are some examples:

. . .with 95% confidence.”

**or**

” . . . 19 times out of 20.”

*Contributed by Anna Spanik, Math teacher, Halifax West High School, Nova Scotia.*