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Chapter 8: Problem 6

A friend is tested to see whether he can tell bottled water from tap water. There are 30 trials (half with bottled water and half with tap water), and he gets 18 right. a. Pick the correct null hypothesis: i. \(\hat{p}=0.50\) ii. \(\hat{p}=0.60\) iii. \(p=0.50\) iv. \(p=0.60\) b. Pick the correct alternative hypothesis: i. \(\hat{p} \neq 0.50\) ii. \(\hat{p}=0.875\) iii. \(p>0.50\) iv. \(p \neq 0.875\)

Short Answer

Expert verified
The correct null hypothesis is \(p=0.50\) and the correct alternative hypothesis is \(p>0.50\).

Step by step solution

01

- Selecting the Correct Null Hypothesis

The null hypothesis states that there is no significant difference and represents a scenario of pure chance. This suggests the expected result, if purely random, would be 50% bottled and 50% tap water. Thus, the null hypothesis is \(p=0.50\) which means the actual probability of guessing it right is 0.50.
02

- Selecting the Correct Alternative Hypothesis

The alternative hypothesis represents the result that would be true if the null hypothesis were false. This means the friend can indeed differentiate between bottled and tap water and his guessing is not just pure chance. So, the alternative hypothesis should be \(p>0.50\), which means the actual probability of guessing it right is more than 0.50.

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Most popular questions from this chapter

In problem \(8.15\) the nutritionist was interested in knowing if the rate of vegetarianism in American adults has increased. She carried out a hypothesis test and found that the observed value of the test statistic was \(2.77 .\) We can calculate that the p-value associated with this is \(0.0028\), which is very close to 0\. Explain the meaning of the p-value in this context. Based on this result, should the nutritionist believe the null hypothesis is true?

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.

The label on a can of mixed nuts says that the mixture contains \(40 \%\) peanuts. After opening a can of nuts and finding 22 peanuts in a can of 50 nuts, a consumer thinks the proportion of peanuts in the mixture differs from \(40 \%\). The consumer writes these hypotheses: \(\mathrm{H}_{0}: \mathrm{p} \neq 0.40\) and \(\mathrm{H}_{\mathrm{a}}: \mathrm{p}=0.44\) where \(p\) represents the proportion of peanuts in all cans of mixed nuts from this company. Are these hypotheses written correctly? Correct any mistakes as needed.

For each of the following, state whether a one-proportion \(z\) -test or a two- proportion z-test would be appropriate, and name the population(s). a. A researcher takes a random sample of voters in western states and voters in southern states to determine if there is a difference in the proportion of voters in these regions who support the death penalty. b. A sociologist takes a random sample of voters to determine if support for the death penalty has changed since 2015 .

A true/false test has 50 questions. Suppose a passing grade is 35 or more correct answers. Test the claim that a student knows more than half of the answers and is not just guessing. Assume the student gets 35 answers correct out of \(50 .\) Use a significance level of \(0.05 .\) Steps 1 and 2 of a hypothesis test procedure are given. Show steps 3 and 4, and be sure to write a clear conclusion. Step $$\text { 1: } \begin{aligned}&\mathrm{H}_{0}: p=0.50 \\\&\mathrm{H}_{\mathrm{a}}: p>0.50\end{aligned}$$ Step 2: Choose the one-proportion \(z\) -test. Sample size is large enough, because \(n p_{0}\) is \(50(0.5)=25\) and \(n\left(1-p_{0}\right)=50(0.50)=25\), and both are more than \(10 .\) Assume the sample is random and \(\alpha=0.05\).
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