/*! 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 64 Give null and alternative hypoth... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 64

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: \(\hat{p}=28 / 40=0.70\) with \(n=40\)

Short Answer

Expert verified
Using StatKey and the given sample data, a p-value is obtained through a Single Proportion Test. Depending on the p-value result, we either reject or fail to reject the null hypothesis of the population proportion being 0.5. The exact p-value depends on the specific randomization distribution generated by StatKey.

Step by step solution

01

State the Null and Alternative Hypotheses

Null hypothesis (H0): The population proportion, p, equals 0.5. Alternative hypothesis (Ha): The population proportion, p, does not equal 0.5.
02

Present the Sample Data

The sample data includes 40 observations, where the sample proportion, \(\hat{p}\), is measured as 0.7 (or 28 out of 40).
03

Use the Single Proportion Test in StatKey

With the sample data and the hypotheses, we can use a Single Proportion Test in StatKey to analyze. For this, 'Edit Data' is used to enter the sample data.
04

Generate a Randomization Distribution and Calculate the p-value

With the sample data entered, StatKey is used to generate a randomization distribution, following the model under the null hypothesis. The p-value is then calculated, indicating the probability of observing the sample proportion given the null hypothesis is true. If the p-value is smaller than the significance level (often 0.05), we reject the null hypothesis in favor of the alternative.

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.

Population Proportion
Let's start by talking about population proportion. In statistics, the population proportion (\( p \) ) is the fraction of the population that has a particular characteristic. Imagine a large bowl of marbles where some are blue and others are red; if you want to find out the proportion of blue marbles, you would look at the fraction referring to the blue ones.
In practice, it's often infeasible to observe the entire population. Therefore, a sample proportion (\( \hat{p} \)) from a smaller, representative sample is used to make inferences about the population. In our exercise, the sample proportion is 0.70, based on 28 favorable observations out of 40 trials. This sample proportion is used to test against our hypotheses concerning the population proportion.
Randomization Distribution
A randomization distribution is an essential step when performing hypothesis tests, especially in determining the validity of our null hypothesis. This distribution shows us what possible sample proportions can look like if the null hypothesis is indeed true.
To build a randomization distribution, we simulate many samples assuming the null hypothesis is correct (\(H_0: p = 0.5\)). These samples help visualize the variability we could expect in the proportion if chance alone dictated the results. The spread of these samples forms the randomization distribution. Each point on this distribution corresponds to a simulated outcome assuming the null hypothesis holds true.
  • This approach allows us to compare our actual sample result (here \(\hat{p} = 0.70\)) to this distribution.

  • It helps determine how unusual or common our sample result is under the assumption that the null hypothesis is true.
p-value Calculation
After creating a randomization distribution, we calculate the p-value, which is crucial to making decisions in hypothesis testing. The p-value represents the probability of observing a sample result as extreme as, or more extreme than, the observed sample proportion, assuming the null hypothesis is true.
In simpler terms, it helps answer: "How surprised should we be to see our sample data if the null hypothesis was true?"
A small p-value indicates that such a result would be rare under the null hypothesis, suggesting that our observed sample proportion (\(\hat{p} = 0.70\)) may not just have happened by random chance.
  • In our context, if the p-value is less than 0.05, we usually conclude that the null hypothesis is unlikely, favoring the alternative hypothesis that the population proportion is not 0.5.

  • The p-value thus helps in determining whether there's enough evidence to reject the null hypothesis.
Being aware of the significance threshold in advance (often 0.05) helps in setting the benchmark for such decisions.

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

Influencing Voters Exercise 4.39 on page 272 describes a possible study to see if there is evidence that a recorded phone call is more effective than a mailed flyer in getting voters to support a certain candidate. The study assumes a significance level of \(\alpha=0.05\) (a) What is the conclusion in the context of thisstudy if the p-value for the test is \(0.027 ?\) (b) In the conclusion in part (a), which type of error are we possibly making: Type I or Type II? Describe what that type of error means in this situation. (c) What is the conclusion if the p-value for the test is \(0.18 ?\)

By some accounts, the first formal hypothesis test to use statistics involved the claim of a lady tasting tea. \({ }^{11}\) In the 1920 's Muriel Bristol- Roach, a British biological scientist, was at a tea party where she claimed to be able to tell whether milk was poured into a cup before or after the tea. R.A. Fisher, an eminent statistician, was also attending the party. As a natural skeptic, Fisher assumed that Muriel had no ability to distinguish whether the milk or tea was poured first, and decided to test her claim. An experiment was designed in which Muriel would be presented with some cups of tea with the milk poured first, and some cups with the tea poured first. (a) In plain English (no symbols), describe the null and alternative hypotheses for this scenario. (b) Let \(p\) be the true proportion of times Muriel can guess correctly. State the null and alternative hypothesis in terms of \(p\).

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 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

Pesticides and ADHD Are children with higher exposure to pesticides more likely to develop ADHD (attention-deficit/hyperactivity disorder)? In one study, authors measured levels of urinary dialkyl phosphate (DAP, a common pesticide) concentrations and ascertained ADHD diagnostic status (Yes/No) for 1139 children who were representative of the general US population. \(^{7}\) The subjects were divided into two groups based on high or low pesticide concentrations, and we compare the proportion with ADHD in each group. (a) Define the relevant parameter(s) and state the null and alternative hypotheses. (b) In the sample, children with high pesticide levels were more likely to be diagnosed with ADHD. Can we necessarily conclude that, in the population, children with high pesticide levels are more likely to be diagnosed with ADHD? (Whether or not we can make this generalization is, in fact, the statistical question of interest.) (c) In the study, evidence was found to support the alternative hypothesis. Explain what that means in the context of pesticide exposure and ADHD?

Using the definition of a p-value, explain why the area in the tail of a randomization distribution is used to compute a p-value.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Probability and 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.