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Chapter 8: Problem 6

Alisha is conducting a paired differences test for a "before \(\left(B\right.\) score) and after \((A \text { score })^{n}\) situation. She is interested in testing whether the average of the "before" scores is higher than that of the "after" scores. (a) To use a right-tailed test, how should Alisha construct the differences between the "before" and "after" scores? (b) To use a left-tailed test, how should she construct the differences between the "before" and "after" scores?

Short Answer

Expert verified
(a) Use \( D = B - A \) for the right-tailed test. (b) Use \( D = A - B \) for the left-tailed test.

Step by step solution

01

Understanding Right-tailed test

To perform a right-tailed test, the alternative hypothesis should be that the mean of the differences is greater than zero. This means that the differences should be calculated as \( D = B - A \). Here, a positive difference \( D \) indicates that the 'before' score is greater than the 'after' score, supporting Alisha's hypothesis that 'before' scores are higher than 'after' scores.
02

Calculating Differences for Right-tailed Test

To construct these differences for the right-tailed test, Alisha should calculate \( D_1 = B_1 - A_1, D_2 = B_2 - A_2, \ldots, D_n = B_n - A_n \), where each \( D_i \) represents the difference for each pair of data. A right-tailed test aims to determine if the mean of these differences is greater than zero.
03

Understanding Left-tailed test

For a left-tailed test, the alternative hypothesis is that the mean of the differences is less than zero. In this case, Alisha should compute the differences as \( D = A - B \). Here, a negative \( D \) suggests 'before' scores are higher than 'after' scores, which aligns with the hypothesis for a left-tailed test.
04

Calculating Differences for Left-tailed Test

To set up the differences for the left-tailed test, Alisha should calculate \( D_1 = A_1 - B_1, D_2 = A_2 - B_2, \ldots, D_n = A_n - B_n \). This setup is used to check if the mean of these differences is less than zero, indicating the 'before' scores are significantly higher.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Right-tailed Test
A right-tailed test in hypothesis testing is used to check if the sample mean is significantly greater than a certain value. In the context of paired differences, we are looking at whether the average difference between two related sets of scores ("before" and "after") is greater than zero. In Alisha's case, she is interested in finding out if the 'before' scores are indeed higher than the 'after' scores.
  • To conduct a right-tailed test, the differences should be computed as \( D = B - A \).
  • A positive test statistic will support the idea that the 'before' scores generally exceed 'after' scores, aligning with her hypothesis.
For each pair of data, Alisha should calculate these differences \( D_1 = B_1 - A_1, D_2 = B_2 - A_2, \ldots, D_n = B_n - A_n \) and then assess if their mean is greater than zero.
Left-tailed Test
Contrary to a right-tailed test, a left-tailed test is appropriate when we suspect that the sample mean is less than a certain value. In a paired differences setting, this means checking if the average difference is less than zero. For Alisha's situation, using a left-tailed test would suggest an interest in verifying if the 'before' scores are perceived as lower than 'after' scores, but since she believes the opposite, this test checks the opposite hypothesis.
  • The calculation for differences here should be \( D = A - B \).
  • If the mean of these differences turns out to be less than zero, it suggests 'before' scores aren't higher, rejecting Alisha's hypothesis.
She should compute differences for each pair: \( D_1 = A_1 - B_1, D_2 = A_2 - B_2, \ldots, D_n = A_n - B_n \). This setup helps check if Alisha's initial belief holds up statistically.
Hypothesis Testing
Hypothesis testing is a statistical method used to make decisions based on data analysis. It involves evaluating two competing hypotheses: the null hypothesis \( H_0 \) and the alternative hypothesis \( H_a \).
  • The null hypothesis, \( H_0 \), usually states that there is no effect or no difference.
  • The alternative hypothesis, \( H_a \), reflects the effect or difference we suspect exists.
In Alisha's study, \( H_0 \) suggests no difference in 'before' and 'after' scores, while \( H_a \) would propose that 'before' scores differ, which is typically greater than 'after' scores in a right-tailed test. The directionality (right-tailed or left-tailed) depends on which hypothesis aligns with the suspected effect.
Statistical Differences
Statistical differences are used to determine if observed data shifts significantly from expected values under a null hypothesis assumption. In paired differences tests like Alisha's, we are focused on whether the linked 'before' and 'after' scores show significant differences on average.
  • Such differences are measured by comparing the mean of paired differences to zero.
  • A significant statistical difference provides evidence against the null hypothesis, supporting the alternative hypothesis.
Determining statistical differences involves calculating the mean of differences and using tests like the t-test to infer if the observed outcomes are due to chance or an actual effect. In Alisha's right-tailed test, focusing on whether these differences are statistically greater than zero helps to validate her belief about score changes.

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Most popular questions from this chapter

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