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Chapter 8: Problem 46

Suppose you are testing someone to see whether he or she can tell butter from margarine when it is spread on toast. You use many bite-sized pieces selected randomly, half from buttered toast and half from toast with margarine. The taster is blindfolded. The null hypothesis is that the taster is just guessing and should get about half right. When you reject the null hypothesis when it is actually true, that is often called the first kind of error. The second kind of error is when the null is false and you fail to reject. Report the first kind of error and the second kind of error.

Short Answer

Expert verified
A Type I error (or the first kind of error) in this experiment would mean that we incorrectly conclude the taster can differentiate between buttered and margarine toast when they actually cannot. A Type II error (or the second kind of error) would mean we incorrectly conclude the taster cannot differentiate between buttered and margarine toast when they actually can.

Step by step solution

01

Understanding Hypothesis Testing

The first step is to understand what hypothesis testing is. This is a statistical method used to make a decision about a population based on sample data. Here, the null hypothesis it that the taster is just guessing and should get about half right.
02

Define First Type of Error (Type I Error)

Next, a type I error, also known as a false positive, occurs when the null hypothesis is true, but is incorrectly rejected. In the context of this experiment, a type I error would mean we mistakenly conclude the taster can differentiate between buttered and margarine toast, when in reality, they are only guessing.
03

Define Second Type of Error (Type II Error)

A type II error, also known as a false negative, takes place when the null hypothesis is false, but is incorrectly failed to be rejected. In the context of this experiment, a type II error would mean we mistakenly conclude the taster cannot differentiate between buttered and margarine toast, when in reality, they can.

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Most popular questions from this chapter

A hospital readmission is an episode when a patient who has been discharged from a hospital is readmitted again within a certain period. Nationally the readmission rate for patients with pneumonia is \(17 \% .\) A hospital was interested in knowing whether their readmission rate for pneumonia was less than the national percentage. They found 11 patients out of 70 treated for pneumonia in a two-month period were readmitted. a. What is \(\hat{p}\), the sample proportion of readmission? b. Write the null and alternative hypotheses. c. Find the value of the test statistic and explain it in context. d. The p-value associated with this test statistic is \(0.39 .\) Explain the meaning of the \(\mathrm{p}\) -value in this context. Based on this result, does the \(\mathrm{p}\) -value indicate the null hypothesis should be doubted?

Give the null and alternative hypotheses for each test, and state whether a one-proportion z-test or a two-proportion z-test would be appropriate. a. You test a person to see whether he can tell tap water from bottled water. You give him 20 sips selected randomly (half from tap water and half from bottled water) and record the proportion he gets correct to test the hypothesis. b. You test a random sample of students at your college who stand on one foot with their eyes closed and determine who can stand for at least 10 seconds, comparing athletes and nonathletes.

St. Louis County is \(24 \%\) African American. Suppose you are looking at jury pools, each with 200 members, in St. Louis County. The null hypothesis is that the probability of an African American being selected into the jury pool is \(24 \%\). a. How many African Americans would you expect on a jury pool of 200 people if the null hypothesis is true? b. Suppose pool A contains 40 African American people out of 200 , and pool B contains 26 African American people out of 200 . Which will have a smaller p-value and why?

An immunologist is testing the hypothesis that the current flu vaccine is less than \(73 \%\) effective against the flu virus. The immunologist is using a \(1 \%\) significance level and these hypotheses: \(\mathrm{H}_{\mathrm{o}}: p=0.73\) and \(\mathrm{H}_{\mathrm{a}}: p<0.73\). Explain what the \(1 \%\) significance level means in context.

Votes for Independents Judging on the basis of experience, a politician claims that \(50 \%\) of voters in Pennsylvania have voted for an independent candidate in past elections. Suppose you surveyed 20 randomly selected people in Pennsylvania, and 12 of them reported having voted for an independent candidate. The null hypothesis is that the overall proportion of voters in Pennsylvania that have voted for an independent candidate is \(50 \%\). What value of the test statistic should you report?
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