Applications of Two-Sample Hypothesis Test
In a two-sample hypothesis test, we compare the population means (proportions) of two populations by collecting two related samples and performing the test to see if we must reject the null hypothesis. Similar to one-sample test, a two-sample hypothesis test can be right tailed, left-tailed, or two-tailed. A two-sample test can be done on a two independent samples or two dependent samples, including the matched pairs sample
NOTICE
Please avoid showing all various types of two-sample hypotheses in your main posting. You should relate your example to two-sample hypotheses by first trying to match your specific claim about comparing two mean values with the claim for one of the following cases and then write the related hypotheses in words similar to the posted hypotheses of that category, based on type of your calim. Also, indicate if the samples are independent or dependent matched pair. Also, discuss how using a two-sample hypothesis test can be helpful in real world.
Two-sample hypotheses with Independent Samples
Example 1
- Right tailed, two-sample hypotheses
“A teacher claims that the average test score for college ACT exam of males is higher than the average score females.”
The samples are independent since the exam performance of an individual does not impact the performance of other participants. We must place this claim under the alternate hypothesis since it does not include equal sign. Also, the word “higher” indicates the test would be right-tailed. Hence,
Null Hypothesis: The average ACT test score is the same as the average test score of females.
Alternate hypothesis (claim): The average ACT test score of males is higher than the average test score of females.
(right-tailed)
- Left tailed, two-sample hypotheses
A teacher claims that the average test score for college ACT exam of males is lower than the average score of females.
The samples are independent since the exam performance an individual does not impact the performance of other participants. We must place this claim under the alternate hypothesis since it does not include equal sign. Also, the word “lower” indicates the test would be left-tailed. Hence,
Null Hypothesis: The average ACT test score is the same as the average test score of females.
Alternate hypothesis (claim): The average ACT test score of males is lower than the average test score of females.
(left-tailed)
- Two-tailed, two-sample hypotheses
A teacher claims that the average test score for college ACT exam of males is different than the average exam score of females.
The samples are independent since the exam performance an individual does not impact the performance of other participants. We must place this claim under the alternate hypothesis since it does not include equal sign. Also, the word “different than” indicates the test would be two-tailed. Hence,
Null Hypothesis: The average ACT test score is the same as the average test score of females.
Alternate hypothesis (claim): The average ACT test score of males is different than the average test score of females.
(Two-tailed)
Example 2
- A retail store manager claims that the average monthly revenue for the first and second quarters of the year are the same.
Notice that the phrase “are the same” indicates an equal sign between the mean values of two populations. So, we must place the claim under the null hypothesis and to write its complement as the alternate hypothesis.
Null Hypothesis (Claim): The average monthly revenue for the first quarter of the year is the same as that of the second quarter.
Alternate Hypothesis: The average monthly revenue for the first quarter of the year is different than the average monthly sales for the second quarter.
This is a two-tailed, two-sample test (based on the alternate hypothesis).
Two-sample hypotheses with Dependent Samples
In case two samples are dependent or related, we perform a two-sample hypothesis test for dependent samples. Two samples are dependent if their performances are interrelated. For example, the temperature in an area before and after a major rainfall during the summer. The temperature is recorded for the same location before and after the event. A well-known method for dependent samples is “Matched Pair samples” method in which each Y-values (from the second sample) is paired with its related X-value (from the first sample). So sample sizes would be equal.
In matched pair method, we use the difference of two population means.
Example
- A meteorologist claims that “The weather is cooler in our region after the rainfall during the summer”. We can re-write this claim as follows:
“The average temperature in our region is lower after the rainfall during the summer”.
This would be two-paired samples. Let the first population and related sample represent the temperature during the summer before the rainfall, and the second population and its related sample represent the temperature during the summer after the rainfall.
The above claim should be placed under the alternate hypothesis since the word “cooler” or “lower than” does not contain an equal sign. This will be a two-paired sample, left-tailed hypothesis test with following hypotheses:
Alternate Hypothesis (claim): The average temperature in our region is lower after the rain fall during the summer”.
Alternate Hypothesis: (left-tailed)
Null Hypothesis: The average temperature in our region is the same after the rain fall during the summer”.
Null Hypothesis:
- “The weather has not changed in our region after the rainfall during this summer”. We can re-write this claim as follows:
“The average temperature in our region is the same after the rain fall during this summer”.
This would be two-paired samples. Let the first population and related sample represent the temperature during the summer before the rainfall, and the second population and its related sample represent the temperature during the summer after the rainfall.
The above claim should be placed under the null hypothesis since the word “has not changed” or “the same” contains an equal sign. This will be a two-paired sample, two-tailed hypothesis test with following hypotheses:
Null Hypothesis (claim): The average temperature in our region is the same after the rain fall during the summer”.
Null Hypothesis:
Alternate Hypothesis: The average temperature in our region has changed after the rain fall during the summer”
Alternate Hypothesis: (Two-tailed)
I hope these few examples about two-sample hypotheses can help you to provide a proper real world example about two-sample hypotheses along with related discussion, including stating your claim to compare two averages or mean values followed by stating the null and alternate hypotheses in words as well as indicating the type of two-sample test for your example, and discussing about using such test results in real world. Please contact me if you need further help. Good luck.