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Chapter 9: Problem 6

Alisha is conducting a paired differences test for a "before \((B\) score) and after \((A\) score \() "\) 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 right-tailed; (b) Use D = A - B for left-tailed.

Step by step solution

01

Understanding Paired Differences

In a paired differences test, we subtract each pair of before (B) and after (A) scores to understand how much change has occurred. For each pair, the difference is calculated using a suitable expression.
02

Construct Differences for Right-Tailed Test

For a right-tailed test, we test if the 'before' scores are greater than the 'after' scores, i.e., we want to see if B > A. To do this, we construct the differences as D = B - A. If these differences are significantly positive, it supports the hypothesis that the 'before' scores are greater than the 'after' scores.
03

Construct Differences for Left-Tailed Test

For a left-tailed test, we test if the 'before' scores are less than the 'after' scores, i.e., we want to see if B < A. To analyze this, we construct the differences as D = A - B. If these differences are significantly positive, it supports the hypothesis that the 'before' scores are less than the 'after' scores.

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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 is a type of hypothesis test where the critical region is on the right side of the probability distribution. This means that we are looking for evidence to support the hypothesis that a parameter is greater than a certain value. In Alisha's exercise, she is interested in testing whether the 'before' scores are higher than the 'after' scores. To construct a right-tailed test in this context, the differences are calculated as \( D = B - A \). Here, if the calculated differences \((D)\) are largely positive, it indicates that the 'before' scores are significantly greater than the 'after' scores. This helps support the right-tailed test hypothesis.
  • Focus on testing if one group is greater.
  • Critical region is at the extreme right of the distribution.
  • Useful for detecting increases or improvements.
Left-Tailed Test
A left-tailed test is the mirror image of the right-tailed test. Here, we place the critical region on the left side of the distribution. This type of test checks if a parameter is less than a specified value. In Alisha's scenario, she would employ a left-tailed test to examine if the 'before' scores are lower than the 'after' scores. The differences here are calculated as \( D = A - B \). If the differences \((D)\) are largely positive, it means the 'after' scores are significantly higher, supporting the hypothesis for the left-tailed test.
  • Checks for decreases or reductions.
  • Critical region lies at the far left.
  • Valuable when investigating declines or negative changes.
Hypothesis Testing
Hypothesis testing is a statistical method that allows us to make decisions about population parameters based on sample data. The process involves formulating two opposing hypotheses: the null hypothesis \( (H_0) \) and the alternative hypothesis \( (H_1) \). Alisha's work sets up a null hypothesis that there is no difference between the 'before' and 'after' scores \( (H_0: \mu_B = \mu_A) \). The alternative hypotheses differ for right and left-tailed tests, reflecting the specific changes she anticipates in the scores.
  • Null Hypothesis \( (H_0) \): Assumes no effect or difference.
  • Alternative Hypothesis \( (H_1) \): Represents the effect or difference.
  • Critical value and p-value are key to decision-making.
Statistical Analysis
Statistical analysis is all about using numbers and data to make sense of phenomena and relationships. Alisha is using statistical analysis to understand the impact of a condition or intervention by comparing the 'before' and 'after' scores. By calculating the differences and using hypothesis testing, Alisha can statistically conclude whether there is a significant change. This process allows her to interpret data patterns, draw conclusions, and make informed decisions based on numerical evidence.
  • Involves organizing, interpreting, and presenting data.
  • Helps in identifying trends and testing theories.
  • Crucial for validating research findings and assumptions.

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Most popular questions from this chapter

Again suppose you are the auditor for a very large corporation. The revenue file contains millions of numbers in a large computer data bank (see Problem 5). You draw a random sample of \(n=228\) numbers from this file and \(r=92\) have a first nonzero digit of \(1 .\) Let \(p\) represent the population proportion of all numbers in the computer file that have a leading digit of 1 . i. Test the claim that \(p\) is more than \(0.301\). Use \(\alpha=0.01\). ii. If \(p\) is in fact larger than \(0.301\), it would seem there are too many numbers in the file with leading 1's. Could this indicate that the books have been "cooked" by artificially lowering numbers in the file? Comment from the point of view of the Internal Revenue Service. Comment from the perspective of the Federal Bureau of Investigation as it looks for "profit skimming" by unscrupulous employees. iii. Comment on the following statement: If we reject the null hypothesis at level of significance \(\alpha\), we have not proved \(H_{0}\) to be false. We can say that the probability is \(\alpha\) that we made a mistake in rejecting \(H_{0} .\) Based on the outcome of the test, would you recommend further investigation before accusing the company of fraud?

Generally speaking, would you say that most people can be trusted? A random sample of \(n_{1}=250\) people in Chicago ages \(18-25\) showed that \(r_{1}=45\) said yes. Another random sample of \(n_{2}=280\) people in Chicago ages \(35-45\) showed that \(r_{2}=71\) said yes (based on information from the National Opinion Research Center, University of Chicago). Does this indicate that the population proportion of trusting people in Chicago is higher for the older group? Use \(\alpha=0.05\).

Would you favor spending more federal tax money on the arts? This question was asked by a research group on behalf of The National Institute (Reference: Painting by Numbers, J. Wypijewski, University of California Press). Of a random sample of \(n_{1}=220\) women, \(r_{1}=\) 59 responded yes. Another random sample of \(n_{2}=175\) men showed that \(r_{2}=\) 56 responded yes. Does this information indicate a difference (either way) between the population proportion of women and the population proportion of men who favor spending more federal tax dollars on the arts? Use \(\alpha=0.05\).

Consider independent random samples from two populations that are normal or approximately normal, or the case in which both sample sizes are at least \(30 .\) Then, if \(\sigma_{1}\) and \(\sigma_{2}\) are unknown but we have reason to believe that \(\sigma_{1}=\sigma_{2}\), we can pool the standard deviations. Using sample sizes \(n_{1}\) and \(n_{2}\), the sample test statistic \(\bar{x}_{1}-\bar{x}_{2}\) has a Student's \(t\) distribution, where \(t=\frac{\bar{x}_{1}-\bar{x}_{2}}{s \sqrt{\frac{1}{n_{1}}+\frac{1}{n_{2}}}}\) with degrees of freedom d.f. \(=n_{1}+n_{2}-2\) and where the pooled standard deviation \(s\) is $$s=\sqrt{\frac{\left(n_{1}-1\right) s_{1}^{2}+\left(n_{2}-1\right) s_{2}^{2}}{n_{1}+n_{2}-2}}$$ Note: With statistical software, select the pooled variance or equal variance options. (a) There are many situations in which we want to compare means from populations having standard deviations that are equal. This method applies even if the standard deviations are known to be only approximately equal (see Section \(11.4\) for methods to test that \(\sigma_{1}=\sigma_{2}\) ). Consider Problem 17 regarding average incidence of fox rabies in two regions. For region I, \(n_{1}=16\), \(\bar{x}_{1}=4.75\), and \(s_{1} \approx 2.82\) and for region II, \(n_{2}=15, \bar{x}_{2} \approx 3.93\), and \(s_{2} \approx\) 2.43. The two sample standard deviations are sufficiently close that we can assume \(\sigma_{1}=\sigma_{2}\). Use the method of pooled standard deviation to redo Problem 17 , where we tested if there was a difference in population mean average incidence of rabies at the \(5 \%\) level of significance. (b) Compare the \(t\) value calculated in part (a) using the pooled standard deviation with the \(t\) value calculated in Problem 17 using the unpooled standard deviation. Compare the degrees of freedom for the sample test statistic. Compare the conclusions.

When testing the difference of means for paired data, what is the null hypothesis?
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