Question

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.

Name: Problem 25, Chapter 8: Understandable Statistics, Concepts and Methods
Uploaded: 2021-11-19T22:16:35-08:00
Duration: 3 min 37 s
Channel: Sheryl Ezze

   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.

Understandable Statistics, Concepts and Methods

Charles Henry Brase,… 12th Edition

Chapter 8, Problem 25

Instant Answer

Solved by Expert Sheryl Ezze

11/19/2021

Step 1

The alternate hypothesis is that the two means are not equal. In mathematical terms, this can be written as: \[H_{0}: \mu_{1}=\mu_{2}\] \[H_{1}: \mu_{1} \neq \mu_{2}\] where $\mu_{1}$ and $\mu_{2}$ are the population means of time lost due to hot tempers and Show more…

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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.

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Key Concepts

Contextual Interpretation

The contextual interpretation involves translating the statistical conclusion back into the framework of the original problem. It explains what rejecting or failing to reject the null hypothesis means in practical terms, providing clarity on the impact or significance of the findings in real-world scenarios.

P-value

The P-value quantifies the probability of observing a test statistic at least as extreme as the one computed from the sample data, assuming that the null hypothesis is true. It is a critical measure in hypothesis testing, as it aids in determining whether the evidence is sufficient to reject the null hypothesis based on the predetermined level of significance.

Decision Rule

The decision rule in hypothesis testing involves comparing the P-value to the level of significance (?). If the P-value is less than ?, the result is considered statistically significant, and the null hypothesis is rejected. Conversely, if the P-value is greater than or equal to ?, the null hypothesis is not rejected.

Test Statistic Calculation

The test statistic is a standardized value calculated from sample data, which measures the degree to which the sample deviates from what is expected under the null hypothesis. For means, this could be a z-score when population standard deviations are known or a t-score when they are estimated from the sample, with the latter requiring appropriate degrees of freedom for accurate assessment.

Level of Significance

The level of significance, typically denoted as ?, is the threshold used to determine whether the observed data provide enough evidence to reject the null hypothesis. It represents the probability of making a Type I error, which is rejecting the null hypothesis when it is true, and governs the criteria for decision-making in hypothesis tests.

Sampling Distribution

The sampling distribution is the probability distribution of a given statistic based on a random sample. In hypothesis testing, understanding the appropriate sampling distribution (such as the z-distribution or t-distribution) is crucial for determining the probability of observing a test statistic under the null hypothesis. The choice depends on factors like sample size and whether the population standard deviation is known.

Null and Alternate Hypotheses

The null hypothesis (H0) is a statement asserting that there is no effect or no difference between the groups or conditions being compared, while the alternate hypothesis (Ha) suggests that there is a significant effect or difference. These hypotheses provide a framework to test claims about population parameters using statistical methods.

Assumptions

Assumptions in hypothesis testing include conditions such as the independence of observations, the normality or approximate normality of the sampling distribution (often guaranteed by the central limit theorem for large samples), and specific distribution shapes like mound-shaped or symmetric distributions for smaller samples. These assumptions underpin the validity of the statistical tests used.

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