Objectives

1.Describe the properties of the Student's distribution

2.Construct confidence intervals for a population mean when the population standard deviation is unknown

Objective 1

Describe the Properties of the Student's Distribution

Student's Distribution

When constructing a confidence interval where we know the population standard deviation , the confidence interval is .The critical value is because the quantity has a normal distribution.

It is rare that we would know the value of while needing to estimate the value of .In practice, it is more common that is unknown.When we don't know the value of , we may replace it with the sample standard deviation .However, we cannot then use as the critical value, because the quantity does not have a normal distribution.The distribution of this quantity is called the Student's distribution.

There are actually many different Student's distributions and they are distinguished by a quantity called the degrees of freedom. When using the Student's distribution to construct a confidence interval for a population mean, the number of degrees of freedom is 1 less than the sample size.

Student's distributions are symmetric and unimodal, just like the normal distribution.However, they are more spread out. The reason is that is, on the average, a bit smaller than . Also, since is random, whereas is constant, replacing with increases the spread. When the number of degrees of freedom is small, the tendency to be more spread out is more pronounced. When the number of degrees of freedom is large, tends to be close to , so the distribution is very close to the normal distribution.

The Critical Value

To find the critical value for a confidence

interval, let 1 be the confidence level.

The critical value is then , because

the area under the Student's t distribution

between and is 1 .

The critical value can be found in Table A.3, in the row corresponding to the number of degrees of freedom and the column corresponding to the desired confidence level or by technology.

Example:A simple random sample of size 10 is drawn from a normal population.Find the critical value for a 95% confidence interval.

Solution:

Degrees of Freedom Not in the Table

If the desired number of degrees of freedom isn't listed in Table A.3, then

If the desired number is less than 200, use the next smaller number that is in the table.

If the desired number is greater than 200, use the -value found in the last row of Table A.3, or use Table A.2.

Assumptions

1.

2.

When the sample size is small ( ), we must _________________________________ _________________________________________________________________________. A simple method is to draw a dotplot or boxplot of the sample.If there are no outliers, and if the sample is not strongly skewed, then it is reasonable to assume the population is approximately normal and it is appropriate to construct a confidence interval using the Student's distribution.

Objective 2

Construct Confidence Intervals for a Population Mean

when the Population Standard Deviation is Unknown

If the assumptions are satisfied, the confidence interval for when is unknown is found using the following steps:

Step 1:Compute the sample mean and sample standard deviation, , if they are not given.

Step 2:Find the number of degrees of freedom - 1 and the critical value .

Step 3:Compute the standard error and multiply it by the critical value to obtain the margin of error .

Step 4:Construct the confidence interval: .

Step 5:Interpret the result.

Example 1:A food chemist analyzed the calorie content for a popular type of chocolate cookie. Following are the numbers of calories in a sample of eight cookies.

113,114,111,116,115,120,118,116

Find a 98% confidence interval for the mean number of calories in this type of cookie.

Solution:

Example 2:A sample of 123 people aged 18-22 reported the number of hours they spent on the Internet in an average week.The sample mean was 8.20 hours, with a sample standard deviation of 9.84 hours.Assume this is a simple random sample from the population of people aged 18-22 in the U.S. Construct a 95% confidence interval for , the population mean number of hours per week spent on the Internet by people aged 18-22 in the U.S.

Solution:

Confidence Intervals on the TI-84 PLUS

The TInterval command constructs confidence intervals when the population standard deviation is unknown.This command is accessed by pressing STAT and highlighting the TESTS menu.

If the summary statistics are given the Stats option should be selected for the input option.

If the raw sample data are given, the Data option should be selected.

YOU SHOULD KNOW ...

The properties of the Student's distribution

Why we must determine whether the sample comes from a population that is approximately normal when the sample size is small ( )

How to construct and interpret confidence intervals for a population mean when the population standard deviation is unknown