t tests:confidence intervals and hypothesis tests

www.ibmaths.com

The t test is used in two situations:

1. When the population variance is unknown.2. When the sample size of the population is small.

It can then be used to find confidence intervals or for a hypothesis test.

Estimating the population variance from a sampleWhen the population variance, , is unknown we use the estimator .

The estimator is defined as:

By using the formula for variance we can use expand the estimator to give:

Example: A sample of 8 students recording the following IQs:

89, 101, 100, 92, 105, 72, 120, 105

Find the unbiased estimator for the variance, for the population from which sample has been drawn.

Answers:Method 1:

Method 2: Using a GDCEnter your numbers into a table. Do a 1-var calculation. Your calculator gives the results as:

or

The t-distributionIn the early 20th century a statistican named W.S.Gosset worked for the Guinness firm in Dublin.

Guinness did not let their employees publish using their own names so Gosset published under the name Student. He called this statistical discovery the ‘Student’s t-distribution’.What Gosset discovered was that the density function of the normal distribution changes for each value of n.

The t distribution looks similar to the normal distribution.The other new feature is that we have degrees of freedom. These are noted by the letter v, and are defined as:v=n-1, where n is the number in the sample.

Where T~t(n-1)

We can define a t distribution as follows:

Reading the table to find critical valuesThis is an extract from the information booklet. It shows the t table values.v = degrees of freedom.

An example: Find the critical t values for a 95% confidence interval with v=8.

A quick diagram helps.

Now look up 0.975, and v=8.

So t = 2.306 and -2.306.

-2.306 2.306

Using the table - quick check

Questions Answers

Find the t-values from your table for the given confidence interval with a sample of n.

1. 90% confidence, n=10

2. 95% confidence, n=7

3. 99% confidence, n=15

Hint: remember that v=n-1

1. -1.833,1.833

2. -2.447,2.447

3. -2.977,2.977

Example of confidence intervals using the t-test.The heights of 8 rugby players from a school in Australia was taken. The heights, in cm, are as follows:

181,178,186,184,189,176,180,182

Find a 95% confidence interval for the mean height of the remaining rugby players in the school.Firstly we are using a t-test because the sample is small (n<30) and we do not know the population variance.

Answer:

Find the mean and standard deviation of the sample. This can be done from a GDC using the Stats list.

Find the t-value from the table. Look up p=0.975 and v=7.

The t-value is 2.365.

Put your data into the formula:

This gives you the answer,

(from the formula sheet)

Example of a hypothesis tests using the t-test.The mean average life-span of a Duraid battery is claimed to be 28 hours.A sample of 6 Duraid batteries has been taken and the life-spans of the 6 batteries are below (in hours).

27.8, 26.1, 28.1, 26.7, 27.1, 27.5

Test, at the 95% level, the claims that the battery life is 28 hours.Answer:Firstly we are using a t-test because the sample is small (n<30) and we do not know the population variance.As this is a hypothesis test write out the null and alternate hypotheses.

Draw a diagram:

Find the t-values. p=0.975,v=5.-2.571 2.571

Find your test statistic. This will be:

From the sample find the statistics needed for the test.

Conclusion: As the test statistic falls within the critical region we must reject the null hypothesis in favour of the alternate.