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7.9 Carrying Out a Test for the Difference of Two Population Means

3 min readโ€ขjanuary 7, 2023

Jed Quiaoit

Jed Quiaoit

Josh Argo

Josh Argo


AP Statisticsย ๐Ÿ“Š

265ย resources
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Once you have determined that the assumptions for the two-sample t-test are met, you can proceed to calculate the test statistic and p-value in order to determine the statistical significance of the difference between the two population means. 2๏ธโƒฃ
To calculate the test statistic, you first need to calculate the difference between the two sample means, and then divide this difference by the standard error of the difference, with standard deviations of the two samples and sample sizes for both samples as variables.
Once you have calculated the test statistic, you can use a t-table or a computer program to determine the p-value, which is the probability that the difference between the two sample means occurred by chance. If the p-value is below a certain threshold (usually 0.05), the difference between the means is considered statistically significant and we can conclude that there is a significant difference between the two population means.

Calculating Test Statistics

The first and necessary aspect of our calculations is calculating our t-score. Since we are dealing with quantitative data (means), we need to find our degrees of freedom first. ๐Ÿ’ฏ

Degrees of Freedom

  • When calculating by hand, we will take the smaller of the two samples and subtract 1. This is the same as we did in Unit 7.5 with 1 sample.
  • When performing the test with technology such as a graphing calculator, the degrees of freedom will be given with the output.

Critical Value (t-score)

To calculate our critical value, we used the typical formula:
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-04-14%20at%202.26.41%20PM.png?alt=media&token=7976f7d5-2ca5-4a0f-91dc-90be8b420ade
To make it more specific for a t-score with the difference of two population means, our formula simplifies to:
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-04-14%20at%202.31.12%20PM.png?alt=media&token=65e60045-ac66-4d9d-91e5-f08b1a2f84f3
This can be found on the Formula Sheet by simplifying the given formulas.

Calculating P-Value

Now that we know our appropriate degrees of freedom and our t-score, we can refer to our Formula Sheet and refer to the appropriate row for our df. Looking across the tow, find the t-score value that is closest to the one you calculated for the t-score. Use the tail probability that most closely coordinates to your t-score. ๐ŸฆŠ
A more exact way of calculating the p-value is to perform a 2 sample t-test in some form of technology such as a graphing calculator. As with any t-procedure, you are given the option of typing in the statistical information or entering in the data in list 1.
Once you enter the test in, the output gives you the t-score, df and p-value for your test. On the AP test, it is essential that you write down ALL 3 of these on your response to receive full credit.
For our green bean example from Unit 7.8, this is what our input would look like:
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-04-14%20at%202.47.33%20PM.png?alt=media&token=a5d4028a-f369-44d7-b2bc-e3320c11f465
And our output would be as follows:
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-04-14%20at%202.49.17%20PM.png?alt=media&token=613ac98b-2103-4c6c-88ac-c745c0e8192d

Testing Statistical Claim

Now that you have the numbers you need, you can check the statistical claim of the null hypothesis. โœ”๏ธ
As with any significance test, we are checking to see if our p is lower than the significance level. If our p is low, we reject the null with convincing evidence of the alternate hypothesis. If the p is not lower than the significance level, we fail to reject the null hypothesis.
Once you make your decision, you should be able to see if in fact there is a difference in your two populations.
For our green bean example, our conclusion would be as follows:
Since our p value is essentially 0 and less than 0.05, we reject our H0. We have convincing evidence that the true mean number of green beans picked from Field A differs from that picked in Field B. ๐Ÿ˜ฒ
I made sure to compare our p-value to our significance level, reject/fail to reject H0, and have evidence/not have evidence of the Ha. Also, my answer is in context of the problem! ๐Ÿ˜„
๐ŸŽฅ Watch: AP Stats - Review of Inference: z and t Procedures
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