/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 109 Suppose we wish to test the hypo... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 109

Suppose we wish to test the hypothesis \(H_{0}: \mu=2\) vs. \(H_{1}: \mu \neq 2 .\) We find a two-sided \(p\) -value of .03 and \(a 95 \%\) Cl for \(\mu\) of \((1.5,4.0) .\) Are these two results possibly compatible? Why or why not?

Short Answer

Expert verified
No, these results are incompatible as the p-value indicates significance while the CI suggests \(H_0\) is plausible.

Step by step solution

01

Understanding the Problem

We need to determine if a two-sided p-value of 0.03 and a 95% confidence interval (CI) for \(\mu\) of (1.5, 4.0) are providing consistent information regarding the hypothesis test about \(\mu\). In this scenario, \(H_0\) is \(\mu=2\) and \(H_1\) is \(\mu eq 2\).
02

Analyzing the P-value

A two-sided p-value of 0.03 suggests that there is a statistically significant difference from \(\mu = 2\) at a significance level of 0.05 (since 0.03 < 0.05), leading us to reject \(H_0\).
03

Interpreting the Confidence Interval

The 95% confidence interval for \(\mu\) is (1.5, 4.0). Since this interval contains the value \(\mu = 2\), it suggests there is no strong evidence to reject \(\mu = 2\) within the interval, implying \(\mu = 2\) is a plausible value.
04

Comparing Results for Compatibility

Despite the p-value indicating statistical significance, the confidence interval including 2 suggests otherwise. These results seem incompatible because the p-value suggests rejecting \(H_0\), while the confidence interval implies \(H_0\) may still be plausible.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Understanding the P-Value
The p-value is a crucial element in hypothesis testing. It helps us decide whether the observed data deviates significantly from the null hypothesis. In our scenario, we have a p-value of 0.03.
This means that there is a 3% probability that the results we observed could happen if the null hypothesis (\(H_0: \mu=2\)) were true.
Because this p-value is lower than the common significance level of 0.05 (or 5%), we consider the results statistically significant.
  • A p-value below 0.05 typically means there is enough evidence to reject the null hypothesis.
  • This suggests the data we have is not consistent with the null hypothesis.
Decoding Confidence Intervals
A confidence interval provides a range of values with which we can be fairly certain the population parameter lies. In this case, the 95% confidence interval is (1.5, 4.0) for \(\mu\).
This range indicates that if we repeat the experiment many times, 95 out of 100 times, the calculated interval will contain the true mean \(\mu\).
Here, the interval is quite wide and includes our null hypothesis value \(\mu = 2\).
  • When an interval includes the hypothesized value, it implies that the value is plausible and we lack strong evidence to reject \(\mu = 2\).
  • Wider intervals suggest more variability in data or a smaller sample size.
Exploring Statistical Significance
Statistical significance is a kind of proof we look for in hypothesis testing to determine if our findings are not due to mere chance. The p-value is often used as a tool to judge this.
When we find that our p-value (0.03) is less than our significance level (0.05), it signals statistical significance, suggesting we reject the null hypothesis.
  • However, statistical significance does not always imply practical significance or importance.
  • The discrepancy between the p-value and the confidence interval results leads us to question the dichotomous nature of statistical significance alone.
This difference can occur due to sample size, variation in data, and the spread of the confidence interval, reminding us that multiple aspects should be considered for sound inference.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The researchers decide to extend the study to a 5 -year period and find that 20 of the 200 people develop a cataract over a 5 -year period. Suppose the expected incidence of cataracts among \(65-\) to 69 -year-olds in the general population is \(5 \%\) over a 5 -year period. Test the hypothesis that the 5-year incidence rate of cataracts is different in the excessive-sunlight-exposure group compared with the general population, and report a \(p\) -value (two-sided).

The mortality experience of 8146 male employees of a research, engineering, and metal-fabrication plant in Tonawanda, New York, was studied from 1946 to 1981 [2]. Potential workplace exposures included welding fumes, cutting oils, asbestos, organic solvents, and environmental ionizing radiation, as a result of waste disposal during the Manhattan Project of World War II. Comparisons were made for specific causes of death between mortality rates in workers and U.S. white-male mortality rates from 1950 to 1978 . Suppose that 17 deaths from cirrhosis of the liver were observed among workers who were hired prior to 1946 and who had worked in the plant for 10 or more years, whereas 6.3 were expected based on U.S. white-male mortality rates. What is the SMR for this group?

Suppose \(\frac{\bar{x}-\mu_{0}}{s / \sqrt{n}}=-1.52\) and a one-sample \(t\) test is performed based on seven subjects. What is the two-tailed \(p\) -value?

What is the 25 th percentile of a \(\chi^{2}\) distribution with 20 degrees of freedom? What symbol is used to denote this value?

An experiment was performed to assess the efficacy of an eye drop in preventing "dry eye." A principal objective measure used to assess efficacy is the tear breakup time (TBUT), which is reduced in people with dry eye and which the researchers hope will be increased after use of the eye drop. In the actual study, participants will be randomized to either active drug or placebo, based on a fairly large sample size. However, a pilot study was first performed based on 14 participants. Under protocol \(A\), the participants had their TBUT measured at baseline and were then instructed to not blink their eyes for 3 seconds, after which time the placebo eye drop was instilled. The TBUT was measured again immediately after instillation as well as at \(5,10,\) and 15 minutes postinstillation, during which time the study participants were in a controlled-chamber environment with low humidity to exacerbate symptoms of dry eye. On 2 other days participants received protocols \(B\) and \(C\). Protocol \(B\) was identical to protocol \(A\) except that participants were told not to blink for 6 seconds prior to drop instillation. Protocol \(\mathrm{C}\) was identical to protocol A except that participants were told not to blink their eyes for 10 seconds prior to drop instillation. Note that the same participants were used for each protocol and that for each protocol data are available for each of two eyes. Also, for each eye, two replicate measurements were provided. The data are available in TEAR.DAT with documentation in TEAR.DOC at www.cengagebrain.com. For each protocol, TBUT (in seconds) was measured (1) at baseline (before drop instillation), (2) immediately after drop instillation, (3) 5 minutes after instillation, (4) 10 minutes after instillation, and (5) 15 minutes after instillation. The standard protocol used in previous clinical studies of TBUT is a 6 -second nonblink interval (protocol B). All the following questions concern protocol B data. For this purpose, average TBUT over two replicates and over both eyes to find a summary value for each time period for each participant. Does the effect of the placebo eye drop change over time after drop instillation? (Hint: Compare mean TBUT at \(5,10,\) and 15 minutes postinstillation vs. mean \(\mathrm{TBUT}\) immediately after drop instillation.)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.