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Chapter 8: Problem 12

An economist is testing the hypothesis that the employment rate for law school graduates is different from \(86.7 \%\). The economist is using a \(5 \%\) significance level and these hypotheses: \(\mathrm{H}_{0}: p=0.867\) and \(\mathrm{H}_{\mathrm{a}}: p \neq 0.867 .\) Explain what the \(5 \%\) significance level means in context.

Short Answer

Expert verified
The \(5 \% \) significance level in this context means there is a \(5 \% \) chance of rejecting the null hypothesis (the employment rate is \(86.7 \% \)) when it's in fact true. In other terms, it's the risk level that you are willing to take that you will reject the null hypothesis when it is true.

Step by step solution

01

Understanding Hypothesis Testing

Hypothesis testing is a statistical method to make decisions or inferences about the population parameter based on sample data. Here, we are checking whether the employment rate for law school graduates is \(86.7 \% \) (null hypothesis) or it is not (alternative hypothesis).
02

Understanding Significance Level

The significance level, also denoted as alpha or \(\alpha \), is the probability of rejecting the null hypothesis when it's true. This is also referred to as a 'Type I error'. For our problem, we have a significance level of \(5 \% \), or \(0.05\), meaning there is a \(5 \% \) risk of concluding that a difference exists when there is no actual difference.
03

Applying Significance Level to the Context

In this context, the \(5 \% \) significance level means that if the null hypothesis is true (i.e. the employment rate is indeed \(86.7 \% \)), there is a \(5 \% \) chance of concluding that the employment rate is different from \(86.7 \% \) based on our sample data. In other words, we are allowing a \(5 \% \) probability of making a mistake in rejecting the null hypothesis.

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Most popular questions from this chapter

A friend claims he can predict the suit of a card drawn from a standard deck of 52 cards. There are four suits and equal numbers of cards in each suit. The parameter, \(p\), is the probability of success, and the null hypothesis is that the friend is just guessing. a. Which is the correct null hypothesis? i. \(p=1 / 4\) ii. \(p=1 / 13\) iii. \(p>1 / 4\) iv. \(p>1 / 13\) b. Which hypothesis best fits the friend's claim? (This is the alternative hypothesis.) i. \(p=1 / 4\) ii. \(p=1 / 13\) iii. \(p>1 / 4\) iv. \(p>1 / 13\)

According to the Brookings Institution, \(50 \%\) of eligible 18 - to 29 -year- old voters voted in the 2016 election. Suppose we were interested in whether the proportion of voters in this age group who voted in the 2018 election was higher. Describe the two types of errors we might make in conducting this hypothesis test.

In 2016 a Harris poll estimated that \(3.3 \%\) of American adults are vegetarian. A nutritionist thinks this rate has increased and will take a random sample of American adults and record whether or not they are vegetarian. State the null and alternative hypotheses in words and in symbols.

When a person stands trial for murder, the jury is instructed to assume that the defendant is innocent. Is this claim of innocence an example of a null hypothesis, or is it an example of an alternative hypothesis?

According to a 2018 survey by Timex reported in Shape magazine, \(73 \%\) of Americans report working out one or more times each week. A nutritionist is interested in whether this percentage has increased. A random sample of 200 Americans found 160 reported working out one or more times each week. Carry out the first two steps of a hypothesis test to determine whether the proportion has increased. Explain how you would fill in the required TI calculator entries for \(p_{0}, x\), and \(n\).
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