Statistics 8 Discussiondebate If Failing To Reject The Null Is The S ✓ Solved

Debate if “failing to reject the null” is the same as “accepting the null.” Support your position with examples of acceptance or rejection of the null. Debate the following statement: "Correlation means causation." Determine whether this statement is true or false, and provide reasoning for your determination, using the Possible Relationships Between Variables table from your textbook. Employees from Company A and Company B both receive annual bonuses. What information would you need to test the claim that the difference in annual bonuses is greater than $100 at the 0.05 level of significance? Write out the hypothesis and explain the testing procedure.

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In the realm of statistics, understanding the nuances between failing to reject the null hypothesis and accepting the null hypothesis is crucial. When researchers conduct hypothesis testing, they operate within a framework where they assess evidence against a specific null hypothesis (H₀). Failing to reject the null does not equate to acceptance; rather, it indicates insufficient evidence to discard the null hypothesis (Mendenhall, Beaver, & Beaver, 2008). For instance, if a researcher finds a p-value of 0.08 when testing whether a new drug is more effective than a placebo, with an alpha level of 0.05, they fail to reject H₀. This does not confirm that the drug is ineffective; it simply suggests that the evidence is not strong enough to prove its efficacy. In contrast, if a different study shows a statistically significant result (p < 0.05), researchers reject H₀, supporting the alternative hypothesis (H_a), which posits that there is an effect or difference (Field, 2013).

Next, let us evaluate the statement, "Correlation means causation." The response to this statement is definitively false. While correlation indicates a relationship between two variables, it does not imply that one variable causes the other (Nisbet, Elder, & Miner, 2018). For example, consider the positive correlation between ice cream sales and drowning incidents during summer months. While both increase in warmer weather, it is incorrect to conclude that higher ice cream sales cause more drownings. Instead, a third factor—hot weather—affects both variables. The Possible Relationships Between Variables table in the textbook elucidates that variables can exhibit correlation due to coincidence, a common cause, or direct causation, contradicting the assumption that correlation innately signifies causation (Moore, Notz, & Fligner, 2013).

Now, turning to the scenario involving Company A and Company B, we need to consider what information is necessary to test the claim that the difference in annual bonuses exceeds $100, using a significance level of 0.05. First, we would establish the null hypothesis (H₀) and the alternative hypothesis (H_a), which could be formulated as follows:

• H₀: μ_A - μ_B ≤ 100 (the difference in bonuses is less than or equal to $100)
• H_a: μ_A - μ_B > 100 (the difference in bonuses is greater than $100)

To proceed with testing this hypothesis, we would require several key pieces of information:

1. The sample means (𝑥̄_A and 𝑥̄_B) for both groups to assess their average annual bonuses.
2. The sample sizes (n_A and n_B) for both Company A and Company B.
3. The standard deviations (s_A and s_B) for each group's bonuses to understand the variability within the samples.
4. The significance level (α), which is given as 0.05 in this case.

Assuming that the assumptions of normality and equal variances are met, we could use a two-sample t-test for this analysis. The t-test will allow us to compare the two means and ascertain whether the observed difference is statistically significant. The formula for the t-statistic in this scenario would be:

T = (𝑥̄_A - 𝑥̄_B - 100) / sqrt[(s_A²/n_A) + (s_B²/n_B)]

We would then compare the computed t-statistic with the critical t-value from the t-distribution table based on our chosen alpha level and the degrees of freedom (df). If the t-statistic exceeds the critical value, we would reject the null hypothesis in favor of the alternative hypothesis, concluding that the difference in annual bonuses is indeed greater than $100.

In summary, the concepts of hypothesis testing, correlation versus causation, and understanding statistical significance are foundational to interpreting data accurately. Researchers must approach these topics meticulously to prevent erroneous conclusions and maintain integrity in their findings.

References

• Field, A. (2013). Discovering Statistics Using SPSS. SAGE Publications.
• Mendenhall, W., Beaver, R. J., & Beaver, B. M. (2008). Introduction to Probability and Statistics. Cengage Learning.
• Moore, D. S., Notz, W. I., & Fligner, M. A. (2013). Statistics. McGraw-Hill.
• Nisbet, R., Elder, J., & Miner, G. (2018). Handbook of Statistical Analysis and Data Mining Applications. Academic Press.
• Lapin, R. (2014). Statistics: A First Course. Pearson Education.
• Stack, S. (2017). Statistical Analysis: A Practical Introduction. CRC Press.
• Bluman, A. G. (2017). Elementary Statistics: A Step by Step Approach. McGraw Hill.
• Triola, M. F. (2018). Elementary Statistics. Pearson.
• Peck, R. J., Olsen, C. W., & Devore, J. L. (2015). Introduction to Statistics and Data Analysis. Cengage Learning.
• Weiss, N. A. (2016). Introductory Statistics. Pearson.