91Ó°ÊÓ

Please provide the following information for Problems. (a) What is the level of significance? State the null and alternate hypotheses. (b) Check Requirements What sampling distribution will you use? What assumptions are you making? Compute the sample test statistic and corresponding \(z\) or \(t\) value as appropriate. (c) Find (or estimate) the \(P\) -value. Sketch the sampling distribution and show the area corresponding to the \(P\) -value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level \(\alpha ?\) (e) Interpret your conclusion in the context of the application. Note: For degrees of freedom \(d . f .\) not in the Student's \(t\) table, use the closest \(d . f .\) that is smaller. In some situations, this choice of \(d . f .\) may increase the \(P\) -value a small amount and therefore produce a slightly more "conservative" answer. Management: Lost Time In her book Red Ink Behaviors, Jean Hollands reports on the assessment of leading Silicon Valley companies regarding a manager's lost time due to inappropriate behavior of employees. Consider the following independent random variables. The first variable \(x_{1}\) measures a manager's hours per week lost due to hot tempers, flaming e-mails, and general unproductive tensions: The variable \(x_{2}\) measures a manager's hours per week lost due to disputes regarding technical workers' superior attitudes that their colleagues are "dumb and dispensable": i. Use a calculator with sample mean and standard deviation keys to verify that \(\bar{x}_{1} \approx 4.86, s_{1} \approx 3.18, \bar{x}_{2}=6.5,\) and \(s_{2} \approx 2.88\) ii. Does the information indicate that the population mean time lost due to hot tempers is different (either way) from population mean time lost due to disputes arising from technical workers' superior attitudes? Use \(\alpha=0.05 .\) Assume that the two lost-time population distributions are mound-shaped and symmetric.

Step by step solution

01

Define Significance Level and Hypotheses

The level of significance is given as \( \alpha = 0.05 \). We need to state the null and alternate hypotheses. The null hypothesis \( H_0 \) asserts that there is no difference between the population means of the time lost due to hot tempers and disputes, which can be written as \( \mu_1 = \mu_2 \). The alternate hypothesis \( H_a \) asserts that there is a difference, \( \mu_1 eq \mu_2 \).

02

Select the Sampling Distribution and Compute Test Statistic

Since we are comparing two independent means and assuming equal variances, we will use the Student's \( t \)-distribution for the test statistic. The assumptions are that both population distributions are approximately normal, as stated. The test statistic \( t \) is computed using:\[t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{s_1^2/n_1 + s_2^2/n_2}}\]Assuming equal sample sizes for simplicity, with \( n_1 = n_2 \), and using the given means and standard deviations, calculate the value of \( t \).

03

Find the P-Value

For the computed \( t \)-statistic, determine the corresponding \( P \)-value using a \( t \)-distribution with appropriate degrees of freedom. Since this is a two-tailed test, the \( P \)-value is the probability that the test statistic is as extreme as, or more extreme than, the observed value.

04

Make a Decision

Compare the \( P \)-value to the significance level \( \alpha = 0.05 \). If the \( P \)-value is less than \( \alpha \), reject the null hypothesis; otherwise, fail to reject the null hypothesis. This decision indicates whether the difference between the two means is statistically significant.

05

Interpret the Conclusion

Based on the decision from Step 4, interpret the result in the context of the problem. If we reject the null hypothesis, it means there is significant evidence to claim a difference in the mean times lost due to hot tempers and disputes regarding technical expertise. If we fail to reject the null hypothesis, there isn't significant evidence to claim a difference.

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Key Concepts

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

Level of Significance

In hypothesis testing, the level of significance, denoted by \( \alpha \), is a crucial component. It represents the threshold or probability of rejecting a true null hypothesis, which is also known as making a Type I error. In simpler terms, it tells us how willing we are to make that mistake. For example, an \( \alpha \) level of 0.05 suggests there is a 5% risk of concluding that a difference exists, when there is, in fact, no actual difference. It's a conservative approach that acts as a filter to differentiate between findings due to actual phenomena and those due to random chance.

• Higher \( \alpha \): Less conservative, more false positives.
• Lower \( \alpha \): More conservative, reduced false positives.

Using \( \alpha = 0.05 \) is a common practice in scientific research, offering a balance between being too permissive and too strict.

Null and Alternate Hypotheses

Formulating the null and alternate hypotheses is the first step in hypothesis testing. The null hypothesis \( H_0 \) represents the default or initial statement that there is no effect or no difference between groups. In our exercise, \( H_0 \) asserts that the population mean time lost due to hot tempers is equal to the time lost due to disputes, expressed as \( \mu_1 = \mu_2 \).On the other hand, the alternate hypothesis \( H_a \) suggests that there is a difference, meaning \( \mu_1 eq \mu_2 \). This is a two-tailed hypothesis since we are looking for any difference, regardless of direction.

• Null Hypothesis \( H_0 \): No difference exists.
• Alternate Hypothesis \( H_a \): A difference exists.

Crafting these hypotheses helps guide the research and clarity in what the test aims to demonstrate.

T-Distribution

The t-distribution is essential when dealing with small sample sizes or unknown population standard deviations. Often, when comparing means of two independent samples, the Student's t-distribution is employed, especially when the sample size is less than 30 for each group.The t-distribution accounts for additional variability by having heavier tails compared to the normal distribution, which means it can accommodate data variability better, making it ideal for small or unknown data variations.Calculating the t-distribution involves the test statistic, formulated as:\[t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}\]where \( \bar{x}_1 \) and \( \bar{x}_2 \) are sample means, \( s_1^2 \) and \( s_2^2 \) are sample variances, and \( n_1 \) and \( n_2 \) are the sample sizes.This test assesses whether the observed difference between means is genuine or a result of random variations.

P-Value

The p-value helps assess the strength of the results obtained from hypothesis testing. It represents the probability of observing data at least as extreme as those observed, assuming the null hypothesis is true.In hypothesis testing:

• If \( P \)-value \(< \alpha \), reject the null hypothesis.
• If \( P \)-value \( \geq \alpha \), fail to reject the null hypothesis.

A smaller p-value indicates more evidence against the null hypothesis, suggesting that the observed data is unlikely under the assumption that \( H_0 \) is true.In our problem, a two-tailed test is considered. Therefore, when determining the p-value through the t-distribution, both tails of the distribution are checked for extremeness. This assessment can help you decide whether the findings are due to genuine effects rather than random chance.