Question

(i) What is the level of significance? State the null and alternate hypotheses. (ii) Check Requirements What sampling distribution will you use? What assumptions are you making? What is the value of the sample test statistic? Compute the corresponding $z$ or $t$ value as appropriate. (iii) Find (or estimate) the $P$ -value. Sketch the sampling distribution and show the area corresponding to the $P$ -value. (iv) Based on your answers in parts (i)-(iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level a? (v) 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.Answers may vary due to rounding. 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 variable $x_{1}$ measures manager's hours per week lost due to hot tempers, flaming e-mails, and general unproductive tensions:$$\begin{array}{llllllll}x_{1}: & 1 & 5 & 8 & 4 & 2 & 4 & 10\end{array}$$.The variable $x_{2}$ measures manager's hours per week lost due to disputes regarding technical workers' superior attitudes that their colleagues are "dumb and dispensable":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 $s_{2} \approx 2.88$(a) Does the information indicate that the population mean time lost due to hot tempers is different (either way) from the 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 symmetrical. (b) Find a $95 \%$ confidence interval for $\mu_{1}-\mu_{2}$. Explain the meaning of the confidence interval in the context of the problem.

Name: Problem 21, Chapter 10: Understandable Statistics Concepts and Methods
Uploaded: 2021-06-29T20:00:37-08:00
Duration: 3 min 41 s
Channel: Jameson Kuper

   (i) What is the level of significance? State the null and alternate hypotheses.
(ii) Check Requirements What sampling distribution will you use? What assumptions are you making? What is the value of the sample test statistic? Compute the corresponding $z$ or $t$ value as appropriate.
(iii) Find (or estimate) the $P$ -value. Sketch the sampling distribution and show the area corresponding to the $P$ -value.
(iv) Based on your answers in parts (i)-(iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level a?
(v) 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.Answers may vary due to rounding. 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 variable $x_{1}$ measures manager's hours per week lost due to hot tempers, flaming e-mails, and general unproductive tensions:$$\begin{array}{llllllll}x_{1}: & 1 & 5 & 8 & 4 & 2 & 4 & 10\end{array}$$.The variable $x_{2}$ measures manager's hours per week lost due to disputes regarding technical workers' superior attitudes that their colleagues are "dumb and dispensable":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 $s_{2} \approx 2.88$(a) Does the information indicate that the population mean time lost due to hot tempers is different (either way) from the 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 symmetrical.
(b) Find a $95 \%$ confidence interval for $\mu_{1}-\mu_{2}$. Explain the meaning of the confidence interval in the context of the problem.

Understandable Statistics Concepts and Methods

Charles Brase,… 12th Edition

Chapter 10, Problem 21

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Solved by Expert Jameson Kuper

06/29/2021

Step 1

The level of significance is $\alpha=0.05$. The null hypothesis $H_{0}$ is that the population mean time lost due to hot tempers is equal to the population mean time lost due to disputes arising from technical workers' superior attitudes, i.e., $\mu_{1}=\mu_{2}$. Show more…

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(i) What is the level of significance? State the null and alternate hypotheses. (ii) Check Requirements What sampling distribution will you use? What assumptions are you making? What is the value of the sample test statistic? Compute the corresponding $z$ or $t$ value as appropriate. (iii) Find (or estimate) the $P$ -value. Sketch the sampling distribution and show the area corresponding to the $P$ -value. (iv) Based on your answers in parts (i)-(iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level a? (v) 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.Answers may vary due to rounding. 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 variable $x_{1}$ measures manager's hours per week lost due to hot tempers, flaming e-mails, and general unproductive tensions:$$\begin{array}{llllllll}x_{1}: & 1 & 5 & 8 & 4 & 2 & 4 & 10\end{array}$$.The variable $x_{2}$ measures manager's hours per week lost due to disputes regarding technical workers' superior attitudes that their colleagues are "dumb and dispensable":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 $s_{2} \approx 2.88$(a) Does the information indicate that the population mean time lost due to hot tempers is different (either way) from the 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 symmetrical. (b) Find a $95 \%$ confidence interval for $\mu_{1}-\mu_{2}$. Explain the meaning of the confidence interval in the context of the problem.

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Key Concepts

Contextual Interpretation

Interpreting the results in context means explaining the statistical findings in terms of the real-world application. The conclusion from the hypothesis test and confidence interval is translated into practical terms, indicating whether the data provide evidence of a meaningful difference or effect in the situation under study.

Level of Significance

This is the predetermined cutoff probability (commonly denoted as ?) used in hypothesis testing to decide whether to reject the null hypothesis. It represents the risk of incorrectly rejecting a true null hypothesis (Type I error) and is compared to the P-value from the test.

Null and Alternate Hypotheses

In hypothesis testing, the null hypothesis (H?) states that there is no effect or difference between the groups or parameters being compared, while the alternate hypothesis (H? or Ha) suggests that there is an effect or difference. These hypotheses form the basis for the testing framework and guide the analysis.

Sampling Distribution and Assumptions

The sampling distribution describes the probability distribution of a statistic (such as the difference between sample means) that is derived from random samples of a population. When sample sizes are small and population variances are unknown, the t-distribution is typically used under the assumption that the data are approximately normally distributed and that the samples are independent.

Test Statistic

The test statistic is a standardized value computed during hypothesis testing that measures the degree of deviation of the sample statistic from its null hypothesis value. For comparisons between two means with unknown variances and small samples, the t-statistic is calculated using the difference between sample means, their standard deviations, and sample sizes.

P-value

The P-value is the probability of obtaining test results at least as extreme as the observed results, assuming that the null hypothesis is true. It quantifies the evidence against the null hypothesis; a small P-value indicates strong evidence for rejecting H?, especially if it is less than the predetermined significance level.

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 or equal to ?, the null hypothesis is rejected in favor of the alternate hypothesis; otherwise, there is insufficient evidence to reject the null hypothesis.

Confidence Intervals

A confidence interval provides a range of plausible values for an unknown population parameter, such as the difference in means, calculated from the sample data. The chosen confidence level (for example, 95%) indicates the probability that the interval would contain the true parameter value if the experiment were repeated many times.

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