In Exercise 2 of Tutorial 4, first we developed a confidence interval for the difference between the mean ages of purchasers and non-purchasers of a particular brand of toothpaste (part a), and then performed a t-test to see whether there was sufficient evidence to conclude that there was a difference in the mean age of purchasers and non-purchasers (part b). Based on the sample variances, in both cases we assumed that the two unknown population variances are different. Let’s check now whether this assumption is supported by the data.Namely, using the same data,a) Estimate the ratio of the two population variances with 95% confidence.b) Can we conclude at the 5% significance level that the population variances differ? What do you conclude if the significance level is increased to 10%?In parts (a) and (b) alike, do the calculations both manually and with R.
Question
In Exercise 2 of Tutorial 4, first we developed a confidence interval for the difference between the mean ages of purchasers and non-purchasers of a particular brand of toothpaste (part a), and then performed a t-test to see whether there was sufficient evidence to conclude that there was a difference in the mean age of purchasers and non-purchasers (part b). Based on the sample variances, in both cases we assumed that the two unknown population variances are different. Let’s check now whether this assumption is supported by the data.Namely, using the same data,a) Estimate the ratio of the two population variances with 95% confidence.b) Can we conclude at the 5% significance level that the population variances differ? What do you conclude if the significance level is increased to 10%?In parts (a) and (b) alike, do the calculations both manually and with R.
Solution
To answer this question, we need to perform an F-test to compare the variances of the two populations. Here's how to do it:
a) Estimate the ratio of the two population variances with 95% confidence.
Step 1: Calculate the sample variances. Let's say the sample variance for purchasers is s1^2 and for non-purchasers is s2^2.
Step 2: Calculate the ratio of the sample variances. This is your point estimate of the ratio of the population variances. The ratio is s1^2 / s2^2.
Step 3: Find the degrees of freedom for each sample. The degrees of freedom for a sample is the sample size minus 1. Let's say the degrees of freedom for purchasers is df1 and for non-purchasers is df2.
Step 4: Look up the F critical values for the 95% confidence level and the degrees of freedom in an F-distribution table. The lower critical value is F(df1, df2, 0.025) and the upper critical value is F(df2, df1, 0.025).
Step 5: Calculate the confidence interval for the ratio of the population variances. The lower limit is (s1^2 / s2^2) / F(df2, df1, 0.025) and the upper limit is (s1^2 / s2^2) * F(df1, df2, 0.025).
b) Can we conclude at the 5% significance level that the population variances differ? What do you conclude if the significance level is increased to 10%?
Step 1: Perform an F-test. The test statistic is the ratio of the sample variances, s1^2 / s2^2.
Step 2: Compare the test statistic to the critical values. If the test statistic is less than the lower critical value or greater than the upper critical value, we reject the null hypothesis that the population variances are equal.
Step 3: If we reject the null hypothesis at the 5% significance level, it means there is sufficient evidence to conclude that the population variances differ. If we do not reject the null hypothesis, we cannot conclude that the variances differ.
Step 4: If the significance level is increased to 10%, the critical values will change. If the test statistic is still outside the new critical values, we can still conclude that the variances differ. If it is now inside the critical values, we cannot conclude that the variances differ.
To perform these calculations in R, you can use the var.test() function for the F-test and the confint() function to calculate the confidence interval for the ratio of the variances.
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