Tests concerning the Difference between Two Means

Example 1

Topic: difference between two brands of candles
Question: is there a difference in burning time between two types of candles, say type A and type B.
Burning times for both types are assumed to be approximately normally distributed.
From both brands a random sample from the production of one day are taken.
Brand A, 10 candles, mean burning time 3.7 hours per candle, standard deviation 0.5 hours per candle.
Brand B, 12 candles, mean burning time 4.2 hours per candle, standard deviation 0.8 hours per candle.

STATE To test whether the mean burning time for the two brands differ, a two sample t-test is used, H₀: \(\mu_A - \mu_B = 0\), H_A: \(\mu_A - \mu_B \ne 0\). Significance level \(\alpha\) = 0.05.

CHECK Conditions:

random samples, yes see text
10% condition, yes total production for one day will be more than 100 candles A and 120 candles B
normal distributed parent population/large sample; yes, parent populations are approx. normally distributed

DO Using the calculator, t = -1.786, p-value = 0.090. Because p-value > \(\alpha\), failed to reject H₀.

CONCLUDE No convincing evidence found for a difference between the mean burning type of the two brands of candles.

Exercise 1

Topic: compare website designs. A web developer compares two different designs for a website for a company. Both designs have the same contents, the difference is in the presentation of the information.
The website can be reached using a advertisement link on other websites.
People who click on this link are randomly lead to one of the two designs.
The developer measures the time a visitor stays at the website.
Sampled data:
Design 1, sample size = 120 visitors, mean time on the site is 56 seconds with a standard deviation of 72 seconds.
Design 2, sample size = 110 visitors, mean time on the site is 49 seconds with a standard deviation of 75 seconds.
Test with an appropriate test whether the average time spend on the website differs for the two designs. Use \(\alpha\) = .10.

Exercise 2

Perform these tests using your calculator.
You don’t have to check conditions. Only write up (1) the H₀ and the H_A hypotheses, (2) the value of the test statistic, (3) the p-value and (4) the conclusion reject or not reject H₀.

H_A: \(\mu_1 > \mu_2\) Sample data: \(n_1 = 45, \bar{x_1} = 120, s_1 = 65; n_2 = 45, \bar{x_2} = 100, s_2 = 80\)
H_A: \(\mu_1 < \mu_2\) Sample data: \(n_1 = 110, \bar{x_1} = 225, s_1 = 75; n_2 = 100, \bar{x_2} = 150, s_2 = 100\)
H_A: \(\mu_1 \ne \mu_2\) Sample data: \(n_1 = 90, \bar{x_1} = 195, s_1 = 85; n_2 = 100, \bar{x_2} = 170, s_2 = 85\)

Exercise 3

In the past the difference between the average burning time for candles type A was significant (\(\alpha\) = 0.05) more than 60 minutes longer than for candles type B.
The producer of type B improved the quality of these type of candles. To test whether the difference in average burning time is still more than 60 minutes two independent samples are taken from one day production of both candles.
Sample data:
Brand A, 10 candles, mean burning time 480 minutes per candle, standard deviation 30 minutes per candle.
Brand B, 12 candles, mean burning time 400 minutes per candle, standard deviation 48 minutes per candle.
Test whether it can be concluded at a 5% significance level that the average burning time of type A candles is more than 60 minutes longer than for type B.
In other words, test \(H_0: \mu_A - \mu_B = 60\) against \(H_A: \mu_A - \mu_B > 60\).

Paired Sample Test

A special case of a two sample test is the paired sample test, also known as correlated two sample test.

Study pp. 586-589 from the text book. As you will have noticed, it is actually a special case of a one-sample t-test.

Some example cases.

Repeated measurements. To improve a certain process the management decides to do an intervention. Group 1: scores on objects before an intervention. Group 2: scores on the same objects after the intervention. A two sample t-test on the average scores is not allowed, because the two groups are not sampled independently. A correct test is the one-sample t-test on the differences between the first score and the second score for each object.
Difference in length between men and women who are married. For this a sample of married couples is chosen at random. It is not correct to perform a t-test on the avergae mean values of the lengths of the men and the lengths of the women in the sample, because these are not independently sampled. A correct test is to notate the difference of the legth of the man and of the woman for each couple and use the one-sample t-test on these differences. Of course it aso possible to construct a CI based on these differences.