/*! 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 70 Exercises 4.67 to 4.70 give a p-... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 70

Exercises 4.67 to 4.70 give a p-value. State the conclusion of the test based on this p-value in terms of " Reject \(H_{0} "\) or "Do not reject \(H_{0} "\) if we use a \(5 \%\) significance level. $$ \text { p-value }=0.1145 $$

Short Answer

Expert verified
Do not reject the null hypothesis \(H_{0}\).

Step by step solution

01

Understand the Concept

The P-value approach involves determining 'likely' or 'unlikely', by determining the probability that a value of a test statistics can happen if the null hypothesis \(H_{0}\) is true. If the p-value is less than the level of significance (which is given as 0.05) we reject the null hypothesis. On the other hand, if the p-value is greater than the level of significance, we fail to reject the null hypothesis.
02

Compare the P-value with the significance level

The given p-value is 0.1145 and the significance level is 0.05. Comparing the two, 0.1145 is greater than 0.05.
03

Make a decision

Since the p-value is greater than the level of significance, we cannot reject the null hypothesis \(H_{0}\).

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 P-Value Interpretation
The p-value is an important concept in statistical hypothesis testing. It helps us understand whether the data we observe is consistent with a given hypothesis. When we calculate a p-value, we assess the probability of obtaining test results at least as extreme as the observed results, under the assumption that the null hypothesis is true.

A small p-value indicates that such extreme results would be quite unusual if the null hypothesis were true. This might lead us to question the null hypothesis. Conversely, a larger p-value suggests that the results are consistent with the null hypothesis, as extreme outcomes wouldn't be unusual in such a scenario.

In our exercise, the p-value given is 0.1145, implying there's an 11.45% chance of obtaining the observed data, or something more extreme, assuming the null hypothesis is true. With this information, we can make informed decisions about the validity of the null hypothesis.
Exploring the Null Hypothesis
The null hypothesis, denoted as \( H_{0} \), is a foundational element in hypothesis testing. It represents a statement of no effect, no difference, or no change. Essentially, it's the hypothesis that the researcher tries to disprove or reject.

In many experiments and studies, the null hypothesis will assert that there is no association between variables or no significant difference between certain groups. It's our default position and acts as a starting point for statistical analysis.

In the context of the given exercise, the null hypothesis might claim that a certain treatment has no effect or that a parameter equals a specific value. The main goal of the test is to evaluate whether the data provide enough evidence to reject this hypothesis in favor of an alternative.
Deciphering the Significance Level
The significance level, often symbolized as \( \alpha \), is a threshold set by the researcher before conducting a hypothesis test. It challenges us to decide how much risk of making a Type I error (wrongly rejecting the true null hypothesis) we are willing to accept.

Common significance levels include 0.05, 0.01, and 0.10. In our exercise, the significance level is 0.05, meaning there is a 5% risk of concluding that a difference exists when there is none.

To make a decision using the significance level, we compare it with the p-value:
  • If the p-value is less than the significance level, we reject the null hypothesis.
  • If the p-value is greater than the significance level, we do not reject the null hypothesis.
For our exercise with a p-value of 0.1145 and a significance level of 0.05, the conclusion is to not reject the null hypothesis, as the p-value exceeds the significance level.

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

In Exercises 4.146 to \(4.149,\) hypotheses for a statistical test are given, followed by several possible confidence intervals for different samples. In each case, use the confidence interval to state a conclusion of the test for that sample and give the significance level used. Hypotheses: \(H_{0}: \mu_{1}=\mu_{2}\) vs \(H_{a}: \mu_{1} \neq \mu_{2} .\) In addition, in each case for which the results are significant, state which group ( 1 or 2 ) has the larger mean. (a) \(95 \%\) confidence interval for \(\mu_{1}-\mu_{2}:\) $$ 0.12 \text { to } 0.54 $$ (b) \(99 \%\) confidence interval for \(\mu_{1}-\mu_{2}\) : -2.1 to 5.4 (c) \(90 \%\) confidence interval for \(\mu_{1}-\mu_{2}\) : -10.8 to -3.7

Determine whether the sets of hypotheses given are valid hypotheses. State whether each set of hypotheses is valid for a statistical test. If not valid, explain why not. (a) \(H_{0}: \rho=0 \quad\) vs \(\quad H_{a}: \rho<0\) (b) \(H_{0}: \hat{p}=0.3 \quad\) vs \(\quad H_{a}: \hat{p} \neq 0.3\) (c) \(H_{0}: \mu_{1} \neq \mu_{2} \quad\) vs \(\quad H_{a}: \mu_{1}=\mu_{2}\) (d) \(H_{0}: p=25 \quad\) vs \(\quad H_{a}: p \neq 25\)

Exercises 4.117 to 4.122 give null and alternative hypotheses for a population proportion, as well as sample results. Use StatKey or other technology to generate a randomization distribution and calculate a p-value. StatKey tip: Use "Test for a Single Proportion" and then "Edit Data" to enter the sample information. Hypotheses: \(H_{0}: p=0.5\) vs \(H_{a}: p \neq 0.5\) Sample data: \(\hat{p}=42 / 100=0.42\) with \(n=100\)

Car Window Skin Cancer? A new study suggests that exposure to UV rays through the car window may increase the risk of skin cancer. \(^{43}\) The study reviewed the records of all 1050 skin cancer patients referred to the St. Louis University Cancer Center in 2004 . Of the 42 patients with melanoma, the cancer occurred on the left side of the body in 31 patients and on the right side in the other 11 . (a) Is this an experiment or an observational study? (b) Of the patients with melanoma, what proportion had the cancer on the left side? (c) A bootstrap \(95 \%\) confidence interval for the proportion of melanomas occurring on the left is 0.579 to \(0.861 .\) Clearly interpret the confidence interval in the context of the problem. (d) Suppose the question of interest is whether melanomas are more likely to occur on the left side than on the right. State the null and alternative hypotheses. (e) Is this a one-tailed or two-tailed test? (f) Use the confidence interval given in part (c) to predict the results of the hypothesis test in part (d). Explain your reasoning. (g) A randomization distribution gives the p-value as 0.003 for testing the hypotheses given in part (d). What is the conclusion of the test in the context of this study? (h) The authors hypothesize that skin cancers are more prevalent on the left because of the sunlight coming in through car windows. (Windows protect against UVB rays but not UVA rays.) Do the data in this study support a conclusion that more melanomas occur on the left side because of increased exposure to sunlight on that side for drivers?

Price and Marketing How influenced are consumers by price and marketing? If something costs more, do our expectations lead us to believe it is better? Because expectations play such a large role in reality, can a product that costs more (but is in reality identical) actually be more effective? Baba Shiv, a neuroeconomist at Stanford, conducted a study \(^{21}\) involving 204 undergraduates. In the study, all students consumed a popular energy drink which claims on its packaging to increase mental acuity. The students were then asked to solve a series of puzzles. The students were charged either regular price \((\$ 1.89)\) for the drink or a discount price \((\$ 0.89)\). The students receiving the discount price were told that they were able to buy the drink at a discount since the drinks had been purchased in bulk. The authors of the study describe the results: "the number of puzzles solved was lower in the reduced-price condition \((M=4.2)\) than in the regular-price condition \((M=5.8) \ldots p<0.0001 . "\) (a) What can you conclude from the study? How strong is the evidence for the conclusion? (b) These results have been replicated in many similar studies. As Jonah Lehrer tells us: "According to Shiv, a kind of placebo effect is at work. Since we expect cheaper goods to be less effective, they generally are less effective, even if they are identical to more expensive products. This is why brand-name aspirin works better than generic aspirin and why Coke tastes better than cheaper colas, even if most consumers can't tell the difference in blind taste tests." 22 Discuss the implications of this research in marketing and pricing.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Geometry

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.