Sample Z C1 C2 C3 C4 C5 C6 C7 C8 Test Of Μ 50 Vs

Sample Z C1 C2 C3 C4 C5 C6 C7 C8 Test Of Μ 50 Vs

Analyze the results of thirty one-sample Z-tests conducted to evaluate whether the population mean \(\mu\) is equal to 50, with an assumed standard deviation of 10. The data includes the number of observations, sample means, standard deviations, standard errors, confidence intervals, test statistics (Z-values), and p-values for each test. Based on these, answer the following questions:

1. (A) How many researchers would reject H₀? That is, how many of them made an incorrect decision?
2. (B) If the level of the test is changed from α = 0.05 to α = 0.001, does this change any of your decisions to reject or not reject H₀?
3. (C) In general, should the number of rejections increase or decrease if α = 0.001 is used instead of α = 0.05?

Paper For Above instruction

The analysis of hypothesis testing outcomes, particularly in the context of assessing whether a population mean equals a specified value, involves understanding the implications of Type I and Type II errors, as well as how significance levels influence decision-making. The dataset provided comprises thirty samples, each subjected to a one-sample Z-test to evaluate if the true mean differs from 50, assuming the population standard deviation is known and fixed at 10.

In hypothesis testing, the null hypothesis (H₀) posits that the population mean \(\mu\) equals 50. The alternative hypothesis (H₁) suggests that \(\mu \neq 50\). Researchers use the Z-statistic to determine whether to reject H₀ based on the p-value and the chosen significance level (α). Using the provided test statistics, we can classify the outcomes based on whether the p-values are below the threshold α.

To answer part (A), we first establish the rejection rule: rejecting H₀ occurs if the p-value is less than α. At α = 0.05, the critical Z-value for a two-tailed test is approximately ±1.96. If the absolute value of the Z-statistic exceeds 1.96, H₀ is rejected. Given the p-values provided, we identify how many are below 0.05, which corresponds to the total number of rejections at this level. The sum of such instances indicates the number of researchers rejecting H₀. Since the total number of tests is 30, counting how many p-values are less than 0.05 gives us the number of rejections, which in this context could potentially include incorrect rejections if the null hypothesis is true for some cases.

Part (B) explores the effect of tightening the significance level to α = 0.001. Since this is a stricter criterion, fewer tests will meet the rejection threshold. For a two-tailed test at α = 0.001, the critical Z-value is approximately ±3.29. Thus, only Z-values with an absolute value greater than 3.29 lead to rejection. Comparing the individual Z-values with this threshold or identifying p-values below 0.001 indicates whether any previous rejections at α = 0.05 would still hold. Usually, only the most extreme Z-values will lead to rejection at this lower α, thus reducing the number of rejections overall.

In part (C), it is understood that decreasing α from 0.05 to 0.001 generally results in a decline in the number of rejections because the criterion for statistical significance becomes more stringent. This lower threshold minimizes Type I errors (incorrect rejections of H₀) but may increase Type II errors (failing to reject H₀ when it is false). Therefore, we expect the total number of rejections to decrease when moving from α = 0.05 to α = 0.001, reflecting a more conservative approach to hypothesis testing. Overall, this illustrates the trade-off between Type I and Type II errors governed by the choice of significance level.

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