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Chapter 4: Problem 5

State the null and alternative hypotheses for the statistical test described. Testing to see if there is evidence that the mean of group \(\mathrm{A}\) is not the same as the mean of group \(\mathrm{B}\).

Short Answer

Expert verified
The null hypothesis (\(H_0\)) is: \(\mu_A = \mu_B\) and the alternative hypothesis (\(H_1\) or \(H_a\)) is: \(\mu_A \neq \mu_B\).

Step by step solution

01

Formulate the Null Hypothesis

The null hypothesis asserts that there's no significant difference in the means of the two groups. Therefore, the null hypothesis (\(H_0\)) can be stated as: The mean of group A equals the mean of group B or, more formally, \(\mu_A = \mu_B\) where \(\mu_A\) and \(\mu_B\) are the respective means of the two groups.
02

Formulate the Alternative Hypothesis

The alternative hypothesis, on the other hand, is what we're trying to discover evidence for. That is, the means of the two groups are not the same. This means the alternative hypothesis (\(H_1\) or \(H_a\)) can be stated as: The mean of group A does not equal the mean of group B or, in other terms, \(\mu_A \neq \mu_B\)

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Key Concepts

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

Null Hypothesis
The null hypothesis, often denoted as \( H_0 \), is a foundational concept in hypothesis testing. It represents the statement we assume to be true unless proven otherwise. In many statistical tests, the null hypothesis posits that there is no effect or no difference. In the context of comparing two group means, the null hypothesis assumes that any observed change is due to random chance.

For example, if we're testing if the mean score of students in group A is the same as in group B, the null hypothesis would state that the mean of group A equals the mean of group B. Mathematically, it is expressed as \( \mu_A = \mu_B \). Here, \( \mu_A \) and \( \mu_B \) represent the mean values for groups A and B, respectively.

Why is the null hypothesis important? It serves as a starting point for statistical testing. When conducting experiments or studies, confirming or rejecting the null hypothesis helps determine if there is enough evidence to support an alternative claim.
Alternative Hypothesis
The alternative hypothesis, indicated as \( H_1 \) or \( H_a \), proposes what we aim to prove. It directly contradicts the null hypothesis and suggests that there is an effect or a significant difference between groups.

In our scenario with group A and group B, the alternative hypothesis posits that the means of these groups are not equal. This can be expressed mathematically as \( \mu_A eq \mu_B \). This indicates that if the null hypothesis holds no merit, there is a statistically significant difference between the mean values of the two groups.

Why do we need an alternative hypothesis? It gives direction to our research, allowing us to focus on detecting genuine effects. When evidence is substantial enough to reject the null hypothesis, the alternative hypothesis steps in to explain these findings.
Mean Comparison
Mean comparison is a statistical technique used to determine if there are differences between the means of two or more groups. Understanding mean differences allows researchers to make informed decisions and draw conclusions about data.

For instance, in the exercise we鈥檙e examining, researchers want to see if the mean of group A differs from the mean of group B. This often involves hypothesis testing, where:
  • The null hypothesis \( (H_0) \) asserts that the two means are equal \( (\mu_A = \mu_B) \).
  • The alternative hypothesis \( (H_1) \) suggests that the means are different \( (\mu_A eq \mu_B) \).

Why is mean comparison important? It鈥檚 essential in areas like product testing, quality control, and medical research to determine which group has a higher or lower average effect. By effectively comparing means, stakeholders can make adjustments, innovate, and improve outcomes based on empirical evidence.

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Most popular questions from this chapter

In a test to see whether there is a difference between males and females in average nasal tip angle, the study indicates that " \(p>0.05\)."

Suppose you want to find out if reading speed is any different between a print book and an e-book. (a) Clearly describe how you might set up an experiment to test this. Give details. (b) Why is a hypothesis test valuable here? What additional information does a hypothesis test give us beyond the descriptive statistics we discuss in Chapter \(2 ?\) (c) Why is a confidence interval valuable here? What additional information does a confidence interval give us beyond the descriptive statistics of Chapter 2 and the results of a hypothesis test described in part (b)? (d) A similar study \(^{53}\) has been conducted, and reports that "the difference between Kindle and the book was significant at the \(p<.01\) level, and the difference between the iPad and the book was marginally significant at \(p=.06 . "\) The report also stated that "the iPad measured at \(6.2 \%\) slower reading speed than the printed book, whereas the Kindle measured at \(10.7 \%\) slower than print. However, the difference between the two devices [iPad and Kindle] was not statistically significant because of the data's fairly high variability." Can you tell from the first quotation which method of reading (print or e-book) was faster in the sample or do you need the second quotation for that? Explain the results in your own words.

Interpreting a P-value In each case, indicate whether the statement is a proper interpretation of what a p-value measures. (a) The probability the null hypothesis \(H_{0}\) is true. (b) The probability that the alternative hypothesis \(H_{a}\) is true. (c) The probability of seeing data as extreme as the sample, when the null hypothesis \(H_{0}\) is true. (d) The probability of making a Type I error if the null hypothesis \(H_{0}\) is true. (e) The probability of making a Type II error if the alternative hypothesis \(H_{a}\) is true.

Indicate whether it is best assessed by using a confidence interval or a hypothesis test or whether statistical inference is not relevant to answer it. (a) What percent of US voters support instituting a national kindergarten through \(12^{\text {th }}\) grade math curriculum? (b) Do basketball players hit a higher proportion of free throws when they are playing at home than when they are playing away? (c) Do a majority of adults riding a bicycle wear a helmet? (d) On average, were the 23 players on the 2014 Canadian Olympic hockey team older than the 23 players on the 2014 US Olympic hockey team?

Influencing Voters: Is a Phone Call More Effective? Suppose, as in Exercise \(4.38,\) that we wish to compare methods of influencing voters to support a particular candidate, but in this case we are specifically interested in testing whether a phone call is more effective than a flyer. Suppose also that our random sample consists of only 200 voters, with 100 chosen at random to get the flyer and the rest getting a phone call. (a) State the null and alternative hypotheses in this situation. (b) Display in a two-way table possible sample results that would offer clear evidence that the phone call is more effective. (c) Display in a two-way table possible sample results that offer no evidence at all that the phone call is more effective. (d) Display in a two-way table possible sample results for which the outcome is not clear: there is some evidence in the sample that the phone call is more effective but it is possibly only due to random chance and likely not strong enough to generalize to the population.
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