/*! 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 75 In Exercises 4.75 and \(4.76,\) ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 75

In Exercises 4.75 and \(4.76,\) match the four p-values with the appropriate conclusion: (a) The evidence against the null hypothesis is significant, but only at the \(10 \%\) level. (b) The evidence against the null and in favor of the alternative is very strong. (c) There is not enough evidence to reject the null hypothesis, even at the \(10 \%\) level. (d) The result is significant at a \(5 \%\) level but not at a \(1 \%\) level. \(\begin{array}{llll}\text { I. } & 0.0875 & \text { II. } & 0.5457\end{array}\) III. 0.0217 IV. 0.00003

Short Answer

Expert verified
Correct assignments are: (a) with I. 0.0875, (b) with IV. 0.00003, (c) with II. 0.5457, and (d) with III. 0.0217.

Step by step solution

01

Match p-value with situation (a)

Situation (a) states that 'The evidence against the null hypothesis is significant, but only at the 10% level'. This means we're looking for a p-value less than 0.10 but presumably more than 0.05, as it does not mention the 5% level. The p-value that fits this scenario is 0.0875 (I).
02

Match p-value with situation (b)

Situation (b) tells us 'The evidence against the null and in favor of the alternative is very strong'. In terms of p-value, this signifies we're looking for a very small p-value, the smallest among the provided, as it reflects a stronger evidence against the null hypothesis. The p-value that fits this scenario is 0.00003 (IV).
03

Match p-value with situation (c)

Situation (c) states 'There isn't enough evidence to reject the null hypothesis, even at the 10% level.' This implies the p-value is more than 0.10. The only p-value that fits this scenario is 0.5457 (II).
04

Match p-value with situation (d)

For situation (d), 'The result is significant at a 5% level but not at a 1% level,' we look for a p-value that's lower than 0.05 but greater than 0.01. The appropriate p-value in this case is 0.0217 (III).

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 critical component in statistical hypothesis testing. It helps us understand the evidence against the null hypothesis. The p-value is a calculation that shows how extreme the results would be, assuming the null hypothesis is true. When we have a smaller p-value, it signifies stronger evidence against the null hypothesis. This means that the data we observed is less likely to have occurred by chance if the null hypothesis is true.
Some key things about p-values:
  • A small p-value (usually less than 0.05) indicates strong evidence against the null hypothesis, leading us to consider the alternative hypothesis.
  • A large p-value suggests that the observed data is consistent with the null hypothesis.
  • A very small p-value (like 0.00003) means there is very strong evidence to reject the null hypothesis.
  • P-values help in decision-making in hypothesis testing, calculating the probability of observing the results assuming the null hypothesis is true.
Understanding the range and implications of different p-values is crucial for drawing meaningful conclusions from a hypothesis test.
The Role of the Null Hypothesis
In hypothesis testing, the null hypothesis is the default or starting assumption. We usually denote it by \(H_0\). The null hypothesis posits that there is no effect, difference, or relationship in the situation being tested. It's a statement of 'no change' or 'no difference', essentially suggesting that any observed effect is purely due to random chance.

Here's how the null hypothesis plays its part:
  • The null hypothesis serves as the initial claim we test against during hypothesis testing. It's the baseline that we aim to challenge or disprove.
  • In most tests, the goal is to gather enough evidence to reject the null hypothesis in favor of an alternative hypothesis, \(H_a\). The alternative might suggest some effect, a difference, or a relationship that opposes \(H_0\).
  • The null hypothesis is more straightforward and conservative, acting as a guard against jumping to erroneous conclusions.
  • Rejecting the null hypothesis in the context of a study gives more credence to the alternative hypothesis.
It's important to note that "not rejecting the null" doesn't confirm it's true; it just means there isn鈥檛 enough evidence to support an alternative hypothesis at that time.
What is the Significance Level?
The significance level, denoted by \(\alpha\), is a threshold that helps determine whether the p-value results are strong enough to reject the null hypothesis. It's a crucial part of hypothesis testing. Essentially, this is the level of risk we are willing to take for making a type I error, where we might incorrectly reject a true null hypothesis.

Important points about the significance level:
  • The significance level is typically set at 0.05 (5%), although it can be adjusted based on the context and field of study, such as 0.01 or 0.10.
  • If the p-value is less than or equal to the significance level, we reject the null hypothesis. This is taken as sufficient evidence to support the alternative hypothesis.
  • Different significance levels provide different thresholds for what we consider statistically significant; a lower \(\alpha\) level demands stronger evidence to reject the null hypothesis.
  • Choosing an appropriate significance level is crucial, as it speaks to the balance between sensitivity and specificity in statistical decision-making.
Think of the significance level as the standard we set to judge the robustness of our test results. A result being statistically significant does not mean it is practically significant, which is another aspect to keep in mind in real-world scenarios.

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

Exercise Hours Introductory statistics students fill out a survey on the first day of class. One of the questions asked is "How many hours of exercise do you typically get each week?" Responses for a sample of 50 students are introduced in Example 3.25 on page 207 and stored in the file ExerciseHours. The summary statistics are shown in the computer output. The mean hours of exercise for the combined sample of 50 students is 10.6 hours per week and the standard deviation is 8.04 . We are interested in whether these sample data provide evidence that the mean number of hours of exercise per week is different between male and female statistics students. Variable Gender N Mean StDev Minimum Maximum \(\begin{array}{lllllll}\text { Exercise } & \text { F } 30 & 9.40 & 7.41 & 0.00 & 34.00\end{array}\) \(\begin{array}{llll}20 & 12.40 & 8.80 & 2,00\end{array}\) Discuss whether or not the methods described below would be appropriate ways to generate randomization samples that are consistent with \(H_{0}: \mu_{F}=\mu_{M}\) vs \(H_{a}: \mu_{F} \neq \mu_{M} .\) Explain your reasoning in each case. (a) Randomly label 30 of the actual exercise values with " \(\mathrm{F}^{\prime \prime}\) for the female group and the remaining 20 exercise values with " \(\mathrm{M} "\) for the males. Compute the difference in the sample means, \(\bar{x}_{F}-\bar{x}_{M}\) (b) Add 1.2 to every female exercise value to give a new mean of 10.6 and subtract 1.8 from each male exercise value to move their mean to 10.6 (and match the females). Sample 30 values (with replacement) from the shifted female values and 20 values (with replacement) from the shifted male values. Compute the difference in the sample means, \(\bar{x}_{F}-\bar{x}_{M}\) - (c) Combine all 50 sample values into one set of data having a mean amount of 10.6 hours. Select 30 values (with replacement) to represent a sample of female exercise hours and 20 values (also with replacement) for a sample of male exercise values. Compute the difference in the sample means, \(\bar{x}_{F}-\bar{x}_{M}\)

Income East and West of the Mississippi For a random sample of households in the US, we record annual household income, whether the location is east or west of the Mississippi River, and number of children. We are interested in determining whether there is a difference in average household income between those east of the Mississippi and those west of the Mississippi. (a) Define the relevant parameter(s) and state the null and alternative hypotheses. (b) What statistic(s) from the sample would we use to estimate the difference? (b) What statistic(s) from the sample would we use to estimate the difference?

Indicate whether the analysis involves a statistical test. If it does involve a statistical test, state the population parameter(s) of interest and the null and alternative hypotheses. Using the complete voting records of a county to see if there is evidence that more than \(50 \%\) of the eligible voters in the county voted in the last election

Eating Breakfast Cereal and Conceiving Boys Newscientist.com ran the headline "Breakfast Cereals Boost Chances of Conceiving Boys," based on an article which found that women who eat breakfast cereal before becoming pregnant are significantly more likely to conceive boys. \({ }^{42}\) The study used a significance level of \(\alpha=0.01\). The researchers kept track of 133 foods and, for each food, tested whether there was a difference in the proportion conceiving boys between women who ate the food and women who didn't. Of all the foods, only breakfast cereal showed a significant difference. (a) If none of the 133 foods actually have an effect on the gender of a conceived child, how many (if any) of the individual tests would you expect to show a significant result just by random chance? Explain. (Hint: Pay attention to the significance level.) (b) Do you think the researchers made a Type I error? Why or why not? (c) Even if you could somehow ascertain that the researchers did not make a Type I error, that is, women who eat breakfast cereals are actually more likely to give birth to boys, should you believe the headline "Breakfast Cereals Boost Chances of Conceiving Boys"? Why or why not?

Two p-values are given. Which one provides the strongest evidence against \(\mathrm{H}_{0} ?\) p-value \(=0.02 \quad\) or \(\quad\) p-value \(=0.0008\)
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Statistics

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.