12.1: Two Variances F Test

Section 0: Introduction to Hypothesis Testing for Two Parameters

The hypothesis testing procedures for testing claims about two population parameters is performed in the same way as the hypothesis testing procedures for one population parameter. As previously, we will check the assumptions, setup the hypotheses and compute the test statistic using a specific formula, and then use the critical value approach or the p-value approach to decide whether to reject or not reject the null hypothesis. Next, we will discuss the nuances of setting up and testing the hypothesis to test the claims about two population parameters. We first introduce a few new terms.

Definition: Independent samples

Independent samples – a pair of samples in which the observations in one sample do not influence the observations in the other.

Definition: Dependent samples

Dependent samples – a pair of samples in which the observations in one sample somehow influence the observations in the other.

Definition: Paired samples

Paired samples – a pair of samples in which the observations are paired in a distinct way.

Consider the following three different ways to pick a sample of ten males and a sample of ten females. It can be done on two different occasions or it can be done simultaneously by picking male-female couples or rotating between males and females. Which of the following samples are independent, dependent, and paired?

1. A sample of 10 males and a sample of 10 females were obtained by randomly picking a male first, then a female second, then another male third, and then another female until the numbers were reached.
2. A sample of 10 males and a sample of 10 females were obtained by randomly picking 10 male-female couples.
3. A sample of 10 males and a sample of 10 females were obtained by picking ten random males on one occasion and then picking ten random females on the other.

Ideally, we want to separate the two samples, so the bottom one is the example of independent samples, the top one is the example of dependent samples. Note that dependency does not have to be explicit, and the middle one is the example of paired samples. Other common examples of paired samples are the samples obtained for the same subjects before and after some event.

We will learn how to test statistical claims that compare a pair of independent parameters of two populations such as:

• Two independent variances
• Two independent proportions
• Two independent means
  • Population standard deviations are known
  • Population standard deviations are not known
    • Can be assumed equal
    • Cannot be assumed equal

We will also learn how to test statistical claims about

• Two paired means

Consider the following examples of statistical claims that involve the parameters about two populations.

• “The average GPA of female students is greater than the average GPA of male students” can be symbolically written as

\(\mu_1>\mu_2\) or \(\mu_1-\mu_2>0\)

• “The percentage of baby-boomers that support the democratic party is less than the percentage of millennials” can be symbolically written as

\(p_1<p_2\) or \(p_1-p_2<0\)

• “The income variation is not the same among those who graduated from college and those who didn’t” can be symbolically written as

\(\sigma^2_1\neq\sigma^2_2\) or \(\frac{\sigma^2_1}{\sigma^2_2}\neq1\)

We can create more interesting statements by replacing 0 and 1 with a number!

• by changing 0 to 0.5 we obtain a slightly different claim

\(\mu_1-\mu_2>0.5\)

which can be interpreted as “the average GPA of female students is at least 0.5 greater than the average GPA of male students”.

• by changing 0 to -0.1 we obtain a slightly different claim

\(p_1-p_2<-0.1\)

which can be interpreted as “the percentage of baby-boomers that support the democratic party is less than the percentage of millennials by 10% or more”.

• by changing 1 to 2 we obtain a slightly different claim

\(\frac{\sigma^2_1}{\sigma^2_2}\neq2\)

which can be interpreted as “the income variation is not twice as large among those who graduated from college and those who didn’t”.

The following table summarizes the null and alternative hypothesis for which we have developed the testing procedures.

Testing Procedure	Two Means	Two Proportions	Two Variances
Null Hypothesis, \(H_0\):	\(\mu_1-\mu_2=\mu_0\)	\(p_1-p_2=p_0\)	\(\frac{\sigma^2_1}{\sigma^2_2}=k\)
Alternative Hypothesis, \(H_a\):	\(\mu_1-\mu_2<\mu_0\)	\(p_1-p_2\neq p_0\)	\(\frac{\sigma^2_1}{\sigma^2_2}>k\)

A few more terms:

Definition: Pooled statistic

A pooled statistic – a statistic computed from two sample as if it was one sample.

For example, for a pooled proportion we have the following formula:

\(\hat{p}_p=\frac{x_1+x_2}{n_1+n_2}\)

If the proportion of males in one class is 9 out of 26 and in the other class, it is 12 out of 24, then the pooled proportion is

\(\hat{p}_p=\frac{9+12}{26+24}=\frac{21}{50}=0.42\)

For example, for a pooled standard deviation we have the following formula:

\(s_p=\sqrt{\frac{(n_1-1)s_1^2+(n_2-1)s^2_2}{n_1+n_2-2}}\)

If sample 1 of size 8 has the sample variance 3.84 and sample 2 of size 16 has the sample variance 6.12 the pooled standard deviation is

\(s_p=\sqrt{\frac{(8-1)\cdot3.84+(16-1)\cdot6.12}{8+16-2}}=2.32\)

Section 1: Two Variances \(F\) Test

Next, we will learn how to apply the Two Variances \(F\) Procedure to test a statistical claim about 2 population variances. Consider the following example.

A pharmaceutical company is about to launch a new manufacturing process in addition to the existing one. The quality control manager believes that the new method results in a different variation in the weights of the capsules. To verify the claim, the samples from each production line were obtained and the results are below:

Sample 1: 99.48 98.79 99.14 97.16 99.88 98.74 100.10

Sample 2: 99.32 99.93 99.86 99.89 99.46 97.67 101.09 100.75 102.03 100.73 101.27

At \(5\%\) significance level, test the claim that the two production methods have different variations. You may assume that the distributions of weights are normal in both production lines.

Note that the sample variances of the samples are \(s^2_1=0.95\) and \(s^2_2=1.39\), and the sizes of the samples are \(n_1=7\) and \(n_2=10\).

Now, let’s identify the statistical claim that needs to be tested:

“variations are different”

The key word “variations” suggests that the claim is about the two variances, so we can symbolically express the claim as

\(\sigma^2_1\neq\sigma^2_2\) or \(\frac{\sigma^2_1}{\sigma^2_2}\neq1\)

Since the claim is about the 2 population variances we will use the 2 Variances \(F\) Procedure. Let’s check if all necessary assumptions are satisfied:

• The samples are simple random and independent
• The populations are normally distributed

We will use the following template to perform the hypothesis testing:

In step 1, we will set up the hypothesis:

Since the claim is in the form of an inequality, therefore it must be set as an alternative hypothesis, therefore the null hypothesis is \(\frac{\sigma^2_1}{\sigma^2_2}=1\) and the test is two-tail, that is:

\(H_0: \frac{\sigma^2_1}{\sigma^2_2}=1\) \(H_a: \frac{\sigma^2_1}{\sigma^2_2}\neq1\)

TT

In step 2, we will identify the significance level:

The significance level can always be found in the text of the problem. In our case it is \(5\%\), thus:

\(\alpha=0.05\)

In step 3, we will find the test statistic using the formula:

\(f_0=\frac{s^2_1}{s^2_2}=\frac{0.95}{1.39}=0.683\)

In step 4, we will perform either the critical value approach or p-value approach to test the claim:

• In critical value approach, we construct the rejection region using the \(F\)-curve with \((dfn=n_1-1=7-1=6, dfd=n_2-1=10-1=9\):

RR: less than \(f_{0.975}=0.181\) or greater than \(f_{0.025}=4.320\)

• In p-value approach, we compute the p-value:

P-Value: \(2P(F<0.683)=0.662\)

In step 5, we will draw the conclusion:

• In the critical value approach, we must check whether the test statistic is in the rejection region or not. Our test statistic is \(0.683\) and it is between the critical values \(0.181\) and \(4.320\), thus it is not in the rejection region.
• In the p-value approach, we must check whether the p-value is less than the significance level or not. Our p-value is \(0.662\) and it is greater than \(\alpha=0.05\).

Both tests suggest that we DO NOT reject the null hypothesis in favor of the alternative.

In step 6, we will interpret the results:

Under \(5\%\) significance level we DO NOT have sufficient evidence to suggest that the two production methods have different variations.

We discussed how to apply the Two Variances \(F\) Procedure to test a statistical claim about two population variances or standard deviations.