Hypothesis Tests for Population Means (College Board AP® Statistics)

Revision Note

Author

Naomi C

Expertise

Maths

One-sample t-test for a mean

What is a one-sample t-test?

A one-sample t-test is used to test whether the population mean, $μ$ , of a normally distributed population has changed
- You use a t-test when the population standard deviation, $σ$ , is unknown
  - This requires using the t-distribution, which is similar to the normal distribution
To try to prove your case, you take a recent random sample of size $n$ from the population and calculate the sample mean, $\bar{x}$
- e.g. you randomly sample 20 cats and calculate their mean weight to be 5.2 kg
  - The sample mean is 5.2 kg

What are the hypotheses for a one-sample t-test?

The null hypothesis, $H_{0}$ , is the assumption that the population mean has not changed
- e.g. $H_{0} :$ The mean weight of all cats in a city is 4.6 kg ( $μ = 4.6$ )
  - It is assumed to be correct, unless evidence proves otherwise

The alternative hypothesis, $H_{a}$ , is how you think the population mean has changed
- e.g. $H_{a} :$ The mean weight of all cats in a city has increased from 4.6 kg ( $μ > 4.6$ )
- Remember that a t-test could be one-tailed or two-tailed, this will affect your alternative hypothesis

Exam Tip

When writing out your hypotheses, always fully define the symbol used for the population parameter in context, e.g. '... where $μ$ is the mean weight of all cats in the city'.

What are the conditions for a one-sample t-test?

When performing a one-sample t-test, you must show that it meets the following conditions:
- Items in the sample (or experiment) must satisfy the independence condition
  - by verifying that data is collected by random sampling
  - or random assignment (in an experiment)
  - and, if sampling without replacement, showing that the sample size is less than 10% of the population size
- The population is approximately normally distributed
  - The distribution needs to be approximately symmetric
  - There should be no outliers
- If the population is very skewed, you can only do a t-test if $n \geq 30$

How do I calculate the standardized test statistic (t-value)?

You need a measure of how far the sample mean is from the population mean
- This is the standardized test statistic (in this case, called the t-value)
The t-value, for the mean is given by:
- $t = \frac{\bar{x} - μ}{\frac{s}{\sqrt{n}}}$
- where $\bar{x}$ is the sample mean, $μ$ is the population mean, $s$ is the sample standard deviation, and $n$ is the sample size
The t-value shows how many standard errors the sample mean is from the population mean

Exam Tip

The formula for the standardized test statistic is given in the exam, $\frac{statistic - parameter}{standard error of the statistic}$ , along with tables of parameters and standard errors.

You will need to apply this correctly to get the t-value.

How do I calculate the p-value?

Work out the t-value
Find the appropriate number of degrees of freedom ('dof')
- For a one-sample t-test this is always, $dof = n - 1$
Using the t-distribution table given to you:
- find the row that corresponds to the dof
- identify the t-value in the row that is closest to the calculated value
- write down the value in the corresponding column header
  - this is the p-value
Note that the p-value from the t-table is for one tail

How do I conclude a hypothesis test?

Conclusions to a hypothesis test need to show two things:
- a decision about the null hypothesis
- an interpretation of this decision in the context of the question
To make the decision, compare the p-value to the significance level
- If $p < α$ then the null hypothesis should be rejected
- If $p > α$ then the null hypothesis should not be rejected
In a two-tailed test, double the p-value and compare this to $α$

Exam Tip

Remember that the test should be interpreted within the context of the question.

Use the same language in your conclusion that is used in the problem, e.g. 'The data provides sufficient evidence that the mean weight of all cats in the city has increased'.

What are the steps for performing a one-sample t-test on a calculator?

When using a calculator to conduct a one-sample t-test, you must still write down all steps of the hypothesis testing process:
- State the null and alternative hypotheses and clearly define your parameter
- Describe the test being used and show that the situation meets the conditions required
- Calculate the t-values and the degrees of freedom
- Calculate the p-value using your calculator
  - select a one-sample t-test and enter the relevant summary statistics or data to generate the p-value
- Compare the p-value to the significance level
- Write down the conclusion to the test and interpret it in the context of the problem

Exam Tip

Even if you perform the one-sample t-test on your calculator, it is still important to show all of your working to demonstrate full understanding. Therefore you should still calculate the t-value and the degrees of freedom.

Worked Example

The IQ of all 800 students at Calculus High can be modeled as a normal distribution with mean 126. The principal decides to play classical music during lunchtimes and suspects that this has caused a change in the average IQ of the students. The principal randomly selects 15 students and asks them to complete an IQ test. The mean score for the sample is 127.1 and the sample standard deviation is 3.8. At the 0.05 level of significance, is the principal's suspicion supported by the test?

Method 1: Using the t-table

Define the population parameter, $μ$

Let $μ$ be the mean IQ of all students at Calculus High

Write the null and alternative hypotheses, this will be a two-tailed test as a change is suspected but an increase or a decrease is not specified

$H_{0} : μ = 126 H_{a} : μ \neq 126$

State the type of test being used and verify the conditions for the test

The correct inference procedure is a one-sample t-test for the population mean at $α = 0.05$

The independence condition is satisfied, as
- the sample of 15 students was selected randomly by the principal
- and the sample size, 15, is less than 10% of the population of the school, 800 (15 < 80), which is required as sampling was conducted without replacement
The distribution of IQ scores in Calculus High is normal
The sample size is small ( $n = 15$ , which is less than 30) and the population standard deviation is unknown, so the t-distribution can be used

Calculate the standardized test statistic, using $t = \frac{\bar{x} - μ}{\frac{s}{\sqrt{n}}}$

$t = \frac{127.1 - 126}{\frac{3.8}{\sqrt{15}}} = 1.12112 . . .$

State the number of degrees of freedom

degrees of freedom = 15 - 1 = 14

Find the p-value from the t-tables

Find the row corresponding to 14 degrees of freedom and identify the t-value that is closest to the calculated t-value

closest t-value = 1.076

corresponding p-value is $p = 0.15$

Double this probability because it is a two-tailed test

$p = 0.15 \cdot 2 = 0.3$

Compare the probability to the significance level and state the conclusion of the test

$\begin{array}{rcl} 0.3 & > & 0.05 \\ p & > & α \end{array}$

$H_{0}$ is not rejected

Interpret the result in the context of the question

We do not have sufficient evidence to support the principal's suspicion that the mean IQ of the students at Calculus High is affected by the playing of classical music at lunchtime

Method 2: Using a calculator

Define the population parameter, $μ$

Let $μ$ be the mean IQ of all students at Calculus High

Write the null and alternative hypotheses, this will be a two-tailed test as a change is suspected but an increase or a decrease is not specified

$H_{0} : μ = 126 H_{a} : μ \neq 126$

State the type of test being used and the conditions for the test

The correct inference procedure is a one-sample t-test for the population mean at $α = 0.05$

The independence condition is satisfied, as
- the sample of 15 students was selected randomly by the principal
- and the sample size, 15, is less than 10% of the population of the school, 800 (15 < 80)
The distribution of IQ scores in Calculus High is normal
The sample size is small ( $n = 15$ , which is less than 30) and the population standard deviation is unknown, so the t-distribution can be used

Even though you do not need to enter the t-value or the degrees of freedom to your calculator, it is still important to show the calculation of them

Calculate the standardized test statistic, using $t = \frac{\bar{x} - μ}{\frac{s}{\sqrt{n}}}$

$t = \frac{127.1 - 126}{\frac{3.8}{\sqrt{15}}} = 1.12112 . . .$

State the number of degrees of freedom

degrees of freedom = 15 - 1 = 14

Write down the parameters for the t-test

$μ = 126 \bar{x} = 127.1 s = 3.8 n = 15$

Enter these into your calculator along with the correct alternative hypothesis and calculate the $p$ -value

$p = 0.28109 . . .$

Compare the probability to the significance level and state the conclusion of the test

$\begin{array}{rcl} 0.28109 . . . & > & 0.05 \\ p & > & α \end{array}$

$H_{0}$ is not rejected

Interpret the result in the context of the question

We do not have sufficient evidence to support the principal's suspicion that the mean IQ of the students at Calculus High is affected by the playing of classical music at lunchtime

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Hypothesis Tests for Population Means (College Board AP® Statistics)

Revision Note

One-sample t-test for a mean

What is a one-sample t-test?

What are the hypotheses for a one-sample t-test?

Exam Tip

What are the conditions for a one-sample t-test?

How do I calculate the standardized test statistic (t-value)?

Exam Tip

How do I calculate the p-value?

How do I conclude a hypothesis test?

Exam Tip

What are the steps for performing a one-sample t-test on a calculator?

Exam Tip

Worked Example

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Author: Naomi C