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Chapter 4: Problem 49

The null and alternative hypotheses for a test are given as well as some information about the actual sample(s) and the statistic that is computed for each randomization sample. Indicate where the randomization distribution will be centered. In addition, indicate whether the test is a left-tail test, a right-tail test, or a twotailed test. Hypotheses: \(H_{0}: p_{1}=p_{2}\) vs \(H_{a}: p_{1}>p_{2}\) Sample: \(\hat{p}_{1}=0.3, n_{1}=20\) and \(\hat{p}_{2}=0.167, n_{2}=12\) Randomization statistic \(=\hat{p}_{1}-\hat{p}_{2}\)

Short Answer

Expert verified
The randomization distribution would be centered at 0, assuming the null hypothesis of no difference in proportions. This is a right-tail test because we are testing for the first proportion being greater than the second one.

Step by step solution

01

Decoding the Hypotheses

Given, null hypothesis (H鈧�): \(p_{1} = p_{2}\), i.e., the two populations have the same proportions. Alternative hypothesis (Ha): \(p_{1}> p_{2}\), i.e., the proportion of the first population is greater than the proportion of the second population.
02

Understanding Sample Data

The sample for population 1 has \(\hat{p}_{1} = 0.3\) and \(n_{1} = 20\), and for population 2 it has \(\hat{p}_{2} = 0.167\) and \(n_{2} = 12\). These values are estimates of population proportions and sample sizes for the two populations.
03

Center of Randomization Distribution

Under the null hypothesis, we assume that the two population proportions are equal. Hence, the randomization distribution of \(\hat{p}_{1}-\hat{p}_{2}\) will be centered at 0 as per the null hypothesis.
04

Test Type identification

As our alternative hypothesis says \(p_{1}> p_{2}\), we are interested in finding evidence of the first proportion being greater than the second one. This means our test is a right-tail test, because we are looking at the extreme right end of the distribution for evidence.

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Key Concepts

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

Null Hypothesis
In hypothesis testing, we start by assuming a statement about a population parameter is true, known as the null hypothesis, symbolized as \(H_0\). In many scenarios, the null hypothesis implies no effect or no difference between groups.For example, if you are comparing two proportions, \(p_1\) and \(p_2\), the null hypothesis might be \(H_0: p_1 = p_2\). This means we assume both populations have the same proportion until we have evidence to the contrary.
  • **Purpose**: The null hypothesis serves as the starting point for statistical testing. It provides a baseline that can be rejected or not rejected based on evidence from a sample.
  • **Center**: In the context of the exercise, under \(H_0\), the randomization distribution is centered at zero, indicating no difference in proportions.
  • **Action**: Rejection of the null hypothesis indicates a significant effect or difference has been found through statistical testing.
Alternative Hypothesis
The alternative hypothesis, denoted \(H_a\), is what we want to prove if the null hypothesis is rejected. It proposes a different scenario than the null, indicating the presence of an effect or difference.For this particular exercise, \(H_a: p_1 > p_2\) suggested that the proportion of the first population, \(p_1\), is greater than that of the second population, \(p_2\).
  • **Purpose**: The alternative hypothesis is essentially what the researcher seeks to support. It reflects the idea that there is an actual effect or difference.
  • **Direction**: This forms the basis for determining the type of test performed, such as one-tailed or two-tailed.
  • **Outcome**: If statistical evidence supports \(H_a\), the null hypothesis is rejected in favor of the alternative.
Right-Tail Test
A right-tail test is a type of statistical test where the area of interest is in the right tail of the distribution. This means we want to find evidence of departures from the null hypothesis that would suggest a value greater than what \(H_0\) proposes.In this exercise, identifying that \(H_a\) is \(p_1 > p_2\) means we are conducting a right-tail test because we are looking for evidence that \(p_1\) exceeds \(p_2\).
  • **Purpose**: Right-tail tests are used when the alternative hypothesis suggests that the parameter of interest is greater than the null hypothesis value.
  • **Applications**: Commonly used when testing for effects like increases or improvements.
  • **Judgment**: If test statistics fall into the critical region on the right, the null hypothesis is rejected.
Understanding these concepts will help you navigate hypothesis testing more effectively, ensuring you can determine the proper type of test and interpret results accordingly.

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Most popular questions from this chapter

Flaxseed and Omega-3 Exercise 4.30 on page 271 describes a company that advertises that its milled flaxseed contains, on average, at least \(3800 \mathrm{mg}\) of ALNA, the primary omega-3 fatty acid in flaxseed, per tablespoon. In each case below, which of the standard significance levels, \(1 \%\) or \(5 \%\) or \(10 \%,\) makes the most sense for that situation? (a) The company plans to conduct a test just to double-check that its claim is correct. The company is eager to find evidence that the average amount per tablespoon is greater than 3800 (their alternative hypothesis), and is not really worried about making a mistake. The test is internal to the company and there are unlikely to be any real consequences either way. (b) Suppose, instead, that a consumer organization plans to conduct a test to see if there is evidence against the claim that the product contains at least \(3800 \mathrm{mg}\) per tablespoon. If the organization finds evidence that the advertising claim is false, it will file a lawsuit against the flaxseed company. The organization wants to be very sure that the evidence is strong, since if the company is sued incorrectly, there could be very serious consequences.

You roll a die 60 times and record the sample proportion of 5 's, and you want to test whether the die is biased to give more 5 's than a fair die would ordinarily give. To find the p-value for your sample data, you create a randomization distribution of proportions of 5 's in many simulated samples of size 60 with a fair die. (a) State the null and alternative hypotheses. (b) Where will the center of the distribution be? Why? (c) Give an example of a sample proportion for which the number of 5 's obtained is less than what you would expect in a fair die. (d) Will your answer to part (c) lie on the left or the right of the center of the randomization distribution? (e) To find the p-value for your answer to part (c), would you look at the left, right, or both tails? (f) For your answer in part (c), can you say anything about the size of the p-value?

Hypotheses for a statistical test are given, followed by several possible confidence intervals for different samples. In each case, use the confidence interval to state a conclusion of the test for that sample and give the significance level used. Hypotheses: \(H_{0}: \mu_{1}=\mu_{2}\) vs \(H_{a}: \mu_{1} \neq \mu_{2} .\) In addition, in each case for which the results are significant, state which group ( 1 or 2 ) has the larger mean. (a) \(95 \%\) confidence interval for \(\mu_{1}-\mu_{2}\) : 0.12 to 0.54 (b) \(99 \%\) confidence interval for \(\mu_{1}-\mu_{2}\) : -2.1 to 5.4 (c) \(90 \%\) confidence interval for \(\mu_{1}-\mu_{2}\) : -10.8 to -3.7

In this exercise, we see that it is possible to use counts instead of proportions in testing a categorical variable. Data 4.7 describes an experiment to investigate the effectiveness of the two drugs desipramine and lithium in the treatment of cocaine addiction. The results of the study are summarized in Table 4.14 on page \(323 .\) The comparison of lithium to the placebo is the subject of Example 4.34 . In this exercise, we test the success of desipramine against a placebo using a different statistic than that used in Example 4.34. Let \(p_{d}\) and \(p_{c}\) be the proportion of patients who relapse in the desipramine group and the control group, respectively. We are testing whether desipramine has a lower relapse rate then a placebo. (a) What are the null and alternative hypotheses? (b) From Table 4.14 we see that 20 of the 24 placebo patients relapsed, while 10 of the 24 desipramine patients relapsed. The observed difference in relapses for our sample is $$\begin{aligned}D &=\text { desipramine relapses }-\text { placebo relapses } \\\&=10-20=-10\end{aligned}$$ If we use this difference in number of relapses as our sample statistic, where will the randomization distribution be centered? Why? (c) If the null hypothesis is true (and desipramine has no effect beyond a placebo), we imagine that the 48 patients have the same relapse behavior regardless of which group they are in. We create the randomization distribution by simulating lots of random assignments of patients to the two groups and computing the difference in number of desipramine minus placebo relapses for each assignment. Describe how you could use index cards to create one simulated sample. How many cards do you need? What will you put on them? What will you do with them?

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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