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Chapter 8: Problem 10

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.

Short Answer

Expert verified
The hypotheses are not correctly formulated in the exercise. The correct null hypothesis should be \(H_0 : p = 0.40\), stating that the percentage of peanuts is indeed the claimed 40%. The correct alternative hypothesis should be \(H_a: p \neq 0.40\), suggesting that the percentage of peanuts is not 40%, which is the consumer's suspicion.

Step by step solution

01

Understanding the Null Hypothesis (H0)

The null hypothesis (H0) is a statement of no effect or no difference and is often the opposite of what we're trying to prove. It's typically based on the 'status quo' or current belief. In this case, the company claims that 40% of the nuts are peanuts, so our null hypothesis should be \(H_0 : p = 0.40\), meaning we start with the assumption that the percentage of peanuts is indeed 40%.
02

Understanding the Alternative Hypothesis (Ha)

The alternative hypothesis (Ha) is what we want to prove. It's the claim or theory that the researcher believes to be true or is interested in testing. In this case, the customer suspects that the actual peanut proportion is not 40%. So the alternative hypothesis should be \(H_a: p \neq 0.40\). This means that the peanut proportion is not equal to 40%, which is what the consumer suspects.
03

Correcting the Hypotheses

As discussed earlier, the null and alternative hypotheses mentioned in the question, \(H_0: p \neq 0.40\) and \(H_a: p = 0.44\), are incorrect. The correct expression of the hypotheses should be \(H_0 : p = 0.40\) and \(H_a: p \neq 0.40\).

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Most popular questions from this chapter

Suppose we are testing people to see whether the rate of use of seat belts has changed from a previous value of \(88 \%\). Suppose that in our random sample of 500 people we see that 450 have the seat belt fastened. Which of the following figures has the correct p-value for testing the hypothesis that the proportion who use seat belts has changed? Explain your choice.

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 \(\alpha=0.05\).

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 polling agency takes a random sample of voters in California to determine if a ballot proposition will pass. b. A researcher asks a random sample of residents from coastal states and a random sample of residents of non-coastal states whether they favor increased offshore oil drilling. The researcher wants to determine if there is a difference in the proportion of residents who support off-shore drilling in the two regions.

In each case. choose whether the appropriate test is a one-proportion \(z\) -test or a two-proportion z-test. Name the population(s). a. A researcher takes a random sample of 4 -year-olds to find out whether girls or boys are more likely to know the alphabet. b. A pollster takes a random sample of all U.S. adult voters to see whether more than \(50 \%\) approve of the performance of the current U.S. president. c. A researcher wants to know whether a new heart medicine reduces the rate of heart attacks compared to an old medicine. d. A pollster takes a poll in Wyoming about homeschooling to find out whether the approval rate for men is equal to the approval rate for women. e. A person is studied to see whether he or she can predict the results of coin flips better than chance alone.

When, in a criminal court, a defendant is found "not guilty," is the court saying with certainty that he or she is innocent? Explain.
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