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Chapter 8: Problem 40

Suppose a poll is taken that shows that 281 out of 500 randomly selected, independent people believe the rich should pay more taxes than they do. Test the hypothesis that a majority (more than \(50 \%\) ) believe the rich should pay more taxes than they do. Use a significance level of \(0.05\).

Short Answer

Expert verified
The provided information allows us to conduct an hypothesis test for the population proportion. By calculating the test statistic and determining the p-value, one can conclude on whether a majority of people believe that the rich should pay more taxes than they currently do, at a significance level of 0.05.

Step by step solution

01

- Setting up the Hypothesis

The null hypothesis states that the population proportion is equal to 0.5, i.e. \(H_0: p = 0.5\). The alternative hypothesis counters that, suggesting that more than 50% of the population believes that the rich should pay more taxes, i.e. \(H_1: p > 0.5\).
02

- Calculating the Test Statistic

The test statistic for a proportion is calculated using the following formula: \(Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}\), where \(\hat{p}\) is the sample proportion, \(p_0\) is the hypothesized proportion in the null hypothesis, and \(n\) is the sample size. Substituting the known values: \(Z = \frac{\frac{281}{500} - 0.5}{\sqrt{\frac{0.5 * 0.5}{500}}}\).
03

- Determining the P-value

For this right-tail test, the p-value is the probability of obtaining a value as extreme as, or more than, the observed test statistic under the null hypothesis. By referring to the standard normal distribution table, we can find the corresponding probability.
04

- Drawing Conclusion

If the p-value is less than the significance level of 0.05, it means that such extreme outcomes are rare under the null hypothesis. As a result, the null hypothesis would be rejected, providing evidence in favor of the alternative hypothesis. If the p-value is greater than 0.05, then it's not statistically significant and the null hypothesis is not rejected.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Population Proportion
Understanding the population proportion is crucial when performing hypothesis testing related to categorical data. It represents the fraction of individuals in a population that exhibit a particular characteristic or attribute. For instance, in the context of our exercise, the population proportion refers to the percentage of people who believe the rich should pay more taxes than they currently do. To estimate this, we usually rely on a sample, and from that sample, we calculate a sample proportion, denoted as \( \hat{p} \), which serves as an approximation of the true population proportion.
Null Hypothesis
The null hypothesis is a statement suggesting that no effect or no difference exists in a particular scenario. It is denoted by \( H_0 \) and is posited for the sole purpose of being challenged by the data we observe. In hypothesis testing, we start by assuming that the null hypothesis is true. In the exercise, \( H_0: p = 0.5 \) suggests that only half of the population believes that the rich should pay more taxes; 'p' stands for the population proportion. It sets a reference point from which we can measure the impact of our sample findings.
Alternative Hypothesis
Opposing the null hypothesis is the alternative hypothesis, denoted as \( H_1 \) or \( H_a \). This hypothesis serves to suggest that there is an effect or difference, and in controlled trials, it is what researchers usually aim to support. For the poll mentioned, \( H_1: p > 0.5 \) indicates the belief that more than 50% of the population thinks that the rich should pay more taxes. This hypothesis is put to the test against the null hypothesis through the collection of evidence in the form of data.
Test Statistic
The test statistic is a standardized value that is calculated from sample data during a hypothesis test. It is designed to measure how far a sample statistic deviates from what is expected if the null hypothesis is true. A test statistic can be used to determine whether to reject the null hypothesis, and it is typically compared to a critical value that corresponds with the chosen significance level. In our exercise, the test statistic 'Z' is calculated using the formula \( Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} \), where \( \hat{p} \) is the sample proportion, \( p_0 \) is the hypothesized population proportion, and 'n' is the size of the sample.
P-value
The p-value is a probability that serves to measure the strength of the evidence against the null hypothesis provided by our sample. It shows us how likely it is to observe a test statistic at least as extreme as the one we computed, assuming the null hypothesis is true. A lower p-value signifies that the observed data is less likely under the null hypothesis, thus lending greater support to the alternative hypothesis. In simpler terms, it quantifies the chance that the results from the study could occur by random variation alone. During our hypothesis test, we compare the p-value to a predetermined significance level to decide whether to reject the null hypothesis.
Significance Level
Lastly, the significance level, often denoted by \( \alpha \), is a threshold chosen by the researcher to decide whether to reject the null hypothesis. It is the probability of rejecting the null hypothesis when it is actually true, known as a Type I error. Common significance levels include 0.05 (5%), 0.01 (1%), and 0.10 (10%). In the exercise, a significance level of 0.05 indicates that we are willing to accept a 5% chance of incorrectly rejecting the null hypothesis. If the p-value is less than \( \alpha \), we conclude that the observed result is statistically significant, and we may reject \( H_0 \) in favor of \( H_1 \).

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Most popular questions from this chapter

Suppose you are testing someone to see whether she or he can tell Coke from Pepsi, and you are using 20 trials, half with Coke and half with Pepsi. The null hypothesis is that the person is guessing. a. About how many should you expect the person to get right under the null hypothesis that the person is guessing? b. Suppose person A gets 13 right out of 20 , and person B gets 18 right out of \(20 .\) Which will have a smaller p-value, and why?

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?

An MCQ test has 75 questions with four options each. Suppose a passing grade is 50 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 50 answers correct out of 75 . 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 \(1: \mathrm{H}_{0}: p=0.25\) \(\mathrm{H}_{\mathrm{a}}: p>0.25\) Step 2: Choose the one-proportion \(z\) -test. Sample size is large enough, because \(n p_{0}\) is \(75(0.25)=18.75\), and \(n\left(1-p_{0}\right)=75(0.75)=56.25\), and both are more than \(10 .\) Assume the sample is random and \(\alpha=0.05\).

A study used nicotine gum to help people quit smoking. The study was placebo- controlled, randomized, and double-blind. Each participant was interviewed after 28 days, and success was defined as being abstinent from cigarettes

A teacher giving a true/false test wants to make sure her students do better than they would if they were simply guessing, so she forms a hypothesis to test this. Her null hypothesis is that a student will get \(50 \%\) of the questions on the exam correct. The alternative hypothesis is that the student is not guessing and should get more than \(50 \%\) in the long run. $$ \begin{aligned} &\mathrm{H}_{0}=p=0.50 \\ &\mathrm{H}_{\mathrm{a}}: p>0.50 \end{aligned} $$ A student gets 30 out of 50 questions, or \(60 \%\), correct. The p-value is \(0.079 .\) Explain the meaning of the p-value in the context of this question.
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