/*! 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 36 Explain why we aren鈥檛 safe car... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 36

Explain why we aren鈥檛 safe carrying out a one-sample z test for the population proportion p. No test A college president says, "99\% of the alumni support my fring of Coach Boggs." You contact an SRS of 200 of the college's \(15,000\) living alumni to test the hypothesis \(H_{0} : p=0.99\)

Short Answer

Expert verified
The sample size is too small when considering the minority (1%); the condition \(n \times (1-p_0) \geq 10\) is not met. Thus, a one-sample z test isn't safe.

Step by step solution

01

Understanding the Hypothesis

We are testing the null hypothesis \(H_{0}: p = 0.99\). This means we are checking if the sample supports the claim that 99% of the alumni support the firing of Coach Boggs.
02

Sample Size and Population Proportion Conditions

Before using a one-sample z test, check if the sample size conditions are met: \(n\times p_0 \geq 10\) and \(n\times (1-p_0) \geq 10\). Here, \(n = 200\), \(p_0 = 0.99\). Calculating, \(200 \times 0.99 = 198\) and \(200 \times 0.01 = 2\).
03

Evaluating the Sample Size Condition

The condition \(n \times (1-p_0) \geq 10\) is not met, as it evaluates to 2, which is much less than 10. This shows the sample size is not large enough to use the normal approximation for the proportion.
04

Conclusion on Z Test Suitability

Since the sample does not satisfy the condition \(n\times (1-p_0) \geq 10\), the normal approximation to the binomial distribution is invalid. Thus, we aren't safe using a one-sample z test for this data.

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.

One-Sample Z Test
A one-sample z test is a statistical method used to determine if the proportion of a single sample matches a hypothesis about the population proportion. It is generally used when you have a large sample size from which to infer about a population. This test compares the sample's observed proportion to a theoretical one. For example, in the context given, the hypothetical value was \(p=0.99\), meaning 99% alumni support. However, it's vital that certain conditions about the sample and population are met for accurate results.
Population Proportion
In statistics, the population proportion \(p\) is a parameter that represents the true proportion of a certain characteristic in the population. For instance, the college president believes that 99% of alumni (p = 0.99) support the firing. The purpose of the hypothesis test is to examine if the sample of 200 alumni reflects this proportion. Estimating population proportion accurately relies on proper sample conditions. If the sample isn't big enough, the calculated proportion might not properly represent the population. This can lead to misguided conclusions.
Sample Size Conditions
For a z test to be valid, the sample size must be sufficiently large. This is checked using the conditions \(n\times p_0 \geq 10\) and \(n\times (1-p_0) \geq 10\). These ensure the distribution of sample proportions is approximately normal. In the exercise, it was calculated that \(n = 200\), \(p_0 = 0.99\), leading to \(n\times p_0 = 198\) and \(n\times (1-p_0) = 2\). Since the second condition isn't satisfied, a one-sample z test can't be used to accurately test the hypothesis. It's essential these conditions are met to provide reliability to the z test results.
Normal Approximation
Normal approximation is a critical concept when employing the one-sample z test. Typically, if the sample size conditions \(n\times p_0 \geq 10\) and \(n\times (1-p_0) \geq 10\) are met, the Binomial distribution can be approximated by a Normal distribution, which simplifies calculations. In cases where these conditions aren't met, as with the exercise's calculation where \(n\times (1-p_0) = 2\), using the Normal approximation is invalid because the sample's distribution doesn't fit a normal curve. Relying on a wrong approximation can lead to false results.
Binomial Distribution
The Binomial distribution is the discrete probability distribution of the number of successes in a sequence of independent trials. Each trial results in a success or failure, based on the probability of success \(p\) in any individual trial. When deciding to use Normal approximation via a one-sample z test, the idea is to simplify the binomial distribution of the sample to a more manageable form. However, if sample size conditions aren't met, as in the exercise, the Binomial distribution remains the ground truth and shouldn't be discarded in favor of invalid simplifications.

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 most important condition for sound conclusions from statistical inference is that (a) the data come from a well-designed random sample or randomized experiment. (b) the population distribution be exactly Normal. (c) the data contain no outliers. (d) the sample size be no more than 10% of the population size. (e) the sample size be at least 30.

Study more! The significance test in Exercise 76 yields a P-value of 0.0622. (a) Describe a Type I and a Type II error in this setting. Which type of error could you have made in Exercise 76? Why? (b) Which of the following changes would give the test a higher power to detect \(\mu=120\) minutes: using \(\alpha=0.01\) or \(\alpha=0.10 ?\) Explain.

We want to be rich In a recent year, 73% of first-year college students responding to a national survey identified 鈥渂eing very well-off financially鈥� as an important personal goal. A state university finds that 132 of an SRS of 200 of its first-year students say that this goal is important. Is there good evidence that the proportion of all first-year students at this university who think being very well-off is important differs from the national value, 73%? Carry out a test at the \(\alpha=\) 0.05 significance level to help answer this question.

Error probabilities You read that a statistical test at the \(\alpha=0.01\) level has probability 0.14 of making a Type II error when a specific alternative is true. What is the power of the test against this alternative?

Growing tomatoes An agricultural field trial compares the yield of two varieties of tomatoes for commercial use. Researchers randomly select 10 Variety A and 10 Variety B tomato plants. Then the researchers divide in half each of 10 small plots of land in different locations. For each plot, a coin toss determines which half of the plot gets a Variety A plant; a Variety B plant goes in the other half. After harvest, they compare the yield in pounds for the plants at each location. The 10 differences (Variety \(A-\) Variety \(B )\) give \(\overline{x}=0.34\) and \(s_{x}=0.83\) A graph of the differences looks roughly symmetric and single- peaked with no outliers. Is there convincing evidence that Variety A has the higher mean yield? Perform a significance test using \(\alpha=0.05\) to answer the question.
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

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.