/*! 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 6 More hypotheses For each of the ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 20: Problem 6

More hypotheses For each of the following, write out the alternative hypothesis, being sure to indicate whether it is one-sided or two-sided. a) Consumer Reports discovered that \(20 \%\) of a certain computer model had warranty problems over the first three months. From a random sample, the manufacturer wants to know if a new model has improved that rate. b) The last time a philanthropic agency requested donations, \(4.75 \%\) of people responded. From a recent pilot mailing, they wonder if that rate has increased. c) A student wants to know if other students on her campus prefer Coke or Pepsi.

Short Answer

Expert verified
a) \( H_a : p < 0.20 \) (one-sided) b) \( H_a : p > 0.0475 \) (one-sided) c) \( H_a : p \neq 0.50 \) (two-sided)

Step by step solution

01

Identify the Null Hypothesis

The null hypothesis ( H_0 ) usually states that there is no effect or no difference. We begin by identifying H_0 for each part: (a) H_0 : p = 0.20 , (b) H_0 : p = 0.0475 , and (c) H_0 : p = 0.50 , assuming no preference between Coke and Pepsi.
02

Formulate the Alternative Hypothesis for Each Scenario

For each scenario, we will determine the alternative hypothesis ( H_a ) and specify whether it is one-sided or two-sided: (a) The manufacturer wants to determine if the new model has improved warranty rates. So, H_a : p < 0.20 , indicating a one-sided test. (b) The agency wants to know if the donation response rate has increased. Thus, H_a : p > 0.0475 , also a one-sided test. (c) The student is trying to determine if there is a preference, without specifying which one is preferred. This leads to H_a : p eq 0.50 , indicating a two-sided test.

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.

Null Hypothesis
The null hypothesis, often denoted as \(H_0\), is a fundamental concept in hypothesis testing. It is the statement that there is no effect or no difference in the situation you are testing. In other words, we presume that any observed differences or changes are solely due to random chance. The null hypothesis serves as a benchmark to see whether the observed data provides enough evidence against it.
Let's look at some examples:
  • For the computer model issue, \(H_0\) is that the proportion of warranty problems remains at 20%, or mathematically, \(H_0: p = 0.20\).
  • In the donation response case, \(H_0\) is that the proportion of people responding remains at 4.75%, or \(H_0: p = 0.0475\).
  • For the preference between Coke and Pepsi, \(H_0\) assumes no preference exists, so \(H_0: p = 0.50\).
By starting with the assumption described by the null hypothesis, we can then use statistical methods to determine if there is sufficient evidence to reject it.
Alternative Hypothesis
The alternative hypothesis, denoted as \(H_a\), is the statement that directly contradicts the null hypothesis. It reflects what you want to prove or find evidence for in your analysis. In practice, if the evidence suggests that the null hypothesis is unlikely, the alternative hypothesis is considered a more plausible explanation for the data.
Here鈥檚 how to set them up:
  • For the improved warranty rate, the alternative hypothesis suggests that the new model reduces the warranty problems, thus \(H_a: p < 0.20\).
  • In the case of increased donations, the alternative hypothesis posits an increase, so \(H_a: p > 0.0475\).
  • When testing if there is a preference between Coke and Pepsi, but without prediction, the alternative hypothesis is \(H_a: p eq 0.50\), indicating a difference one way or another.
The choice of alternative hypothesis depends on what you want to test, whether it's a greater than, less than, or different scenario.
One-Sided Test
A one-sided test is a type of hypothesis test where the alternative hypothesis is directional. It means that you are only interested in whether something is either greater than or less than a certain value, not both. This kind of testing is used when the research question or objective clearly points in one direction.To understand this better, look at these scenarios:
  • When checking if the new computer model has a lower rate of warranty issues compared to the initial 20%, you're conducting a one-sided test with an alternative hypothesis of \(H_a: p < 0.20\).
  • For the donations case, we're interested in whether more people responded. Thus, a one-sided test is set with \(H_a: p > 0.0475\).
In both cases, we're looking at an improvement or increase, but not at the other possible direction. This makes it a one-sided hypothesis test.
Two-Sided Test
A two-sided test is a type of hypothesis test where the alternative hypothesis does not state a specific direction. Instead, it checks for any difference from the value stated in the null hypothesis, whether it's more or less. This type of test is appropriate when a deviation is of interest in either direction.The scenario involving preferences for Coke or Pepsi fits this model:
  • The student wants to know if there's a preference one way or the other. They don't specify which drink is or isn't preferred, meaning any difference would be notable. Hence, the alternative hypothesis here is \(H_a: p eq 0.50\).
Here, we allow for deviations on both sides of the value, making it a two-sided test. This ensures that we catch any significant difference, regardless of direction.

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

Parameters and hypotheses For each of the following situations, define the parameter (proportion or mean) and write the null and alternative hypotheses in terms of parameter values. Example: We want to know if the proportion of up days in the stock market is \(50 \%\). Answer: Let \(p=\) the proportion of up days. \(\mathrm{H}_{\mathrm{o}}: p=0.5\) vs. \(\mathrm{H}_{\mathrm{A}}: p \neq 0.5.\) a) A casino wants to know if their slot machine really delivers the 1 in 100 win rate that it claims. b) A pharmaceutical company wonders if their new drug has a cure rate different from the \(30 \%\) reported by the placebo. c) A bank wants to know if the percentage of customers using their website has changed from the \(40 \%\) that used it before their system crashed last week.

Errors For each of the following situations, state whether a Type I, a Type II, or neither error has been made. Explain briefly. a) A bank wants to see if the enrollment on their website is above \(30 \%\) based on a small sample of customers. It tests \(\mathrm{H}_{0}: p=0.3\) vs. \(\mathrm{H}_{\mathrm{A}}: p>0.3\) and rejects the null hypothesis. Later the bank finds out that actually \(28 \%\) of all customers enrolled. b) A student tests 100 students to determine whether other students on her campus prefer Coke or Pepsi and finds no evidence that preference for Coke is not \(0.5 .\) Later, a marketing company tests all students on campus and finds no difference. c) A pharmaceutical company tests whether a drug lifts the headache relief rate from the \(25 \%\) achieved by the placebo. The test fails to reject the null hypothesis because the P-value is 0.465. Further testing shows that the drug actually relieves headaches in \(38 \%\) of people.

More \(P\) -values Which of the following are true? If false, explain briefly. a) A very low \(P\) -value provides evidence against the null hypothesis. b) A high P-value is strong evidence in favor of the null hypothesis. c) A P-value above 0.10 shows that the null hypothesis is true. d) If the null hypothesis is true, you can't get a P-value below 0.01.

Pottery An artist experimenting with clay to create pottery with a special texture has been experiencing difficulty with these special pieces. About \(40 \%\) break in the kiln during firing. Hoping to solve this problem, she buys some more expensive clay from another supplier. She plans to make and fire 10 pieces and will decide to use the new clay if at most one of them breaks. a) Suppose the new, expensive clay really is no better than her usual clay. What's the probability that this test convinces her to use it anyway? (Hint: Use a Binomial model.) b) If she decides to switch to the new clay and it is no better, what kind of error did she commit? c) If the new clay really can reduce breakage to only \(20 \%,\) what's the probability that her test will not detect the improvement? d) How can she improve the power of her test? Offer at least two suggestions.

Loans Before lending someone money, banks must decide whether they believe the applicant will repay the loan. One strategy used is a point system. Loan officers assess information about the applicant, totaling points they award for the person's income level, credit history, current debt burden, and so on. The higher the point total, the more convinced the bank is that it's safe to make the loan. Any applicant with a lower point total than a certain cutoff score is denied a loan. We can think of this decision as a hypothesis test. Since the bank makes its profit from the interest collected on repaid loans, their null hypothesis is that the applicant will repay the loan and therefore should get the money. Only if the person's score falls below the minimum cutoff will the bank reject the null and deny the loan. This system is reasonably reliable, but, of course, sometimes there are mistakes. a) When a person defaults on a loan, which type of error did the bank make? b) Which kind of error is it when the bank misses an opportunity to make a loan to someone who would have repaid it? c) Suppose the bank decides to lower the cutoff score from 250 points to \(200 .\) Is that analogous to choosing a higher or lower value of \(\alpha\) for a hypothesis test? Explain. d) What impact does this change in the cutoff value have on the chance of each type of error?
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Pure 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.