Chapter 4: Problem 88
Using the \(\mathrm{p}\) -value given, are the results significant at a \(10 \%\) level? At a \(5 \%\) level? At a \(1 \%\) level? p-value \(=0.0320\)
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Exercises 4.29 on page 271 and 4.76 on page 287 describe a historical scenario in which a British woman, Muriel BristolRoach, claimed to be able to tell whether milk had been poured into a cup before or after the tea. An experiment was conducted in which Muriel was presented with 8 cups of tea, and asked to guess whether the milk or tea was poured first. Our null hypothesis \(\left(H_{0}\right)\) is that Muriel has no ability to tell whether the milk was poured first. We would like to create a randomization distribution for \(\hat{p},\) the proportion of cups out of 8 that Muriel guesses correctly under \(H_{0}\). Describe a possible approach to generate randomization samples for each of the following scenarios: (a) Muriel does not know beforehand how many cups have milk poured first. (b) Muriel knows that 4 cups will have milk poured first and 4 will have tea poured first.
Give null and alternative hypotheses for a population proportion, as well as sample results. Use StatKey or other technology to generate a randomization distribution and calculate a p-value. StatKey tip: Use "Test for a Single Proportion" and then "Edit Data" to enter the sample information. Hypotheses: \(H_{0}: p=0.5\) vs \(H_{a}: p \neq 0.5\) Sample data: \(\hat{p}=28 / 40=0.70\) with \(n=40\)
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?
Flying Home for the Holidays, On Time In Exercise 4.115 on page \(302,\) we compared the average difference between actual and scheduled arrival times for December flights on two major airlines: Delta and United. Suppose now that we are only interested in the proportion of flights arriving more than 30 minutes after the scheduled time. Of the 1,000 Delta flights, 67 arrived more than 30 minutes late, and of the 1,000 United flights, 160 arrived more than 30 minutes late. We are testing to see if this provides evidence to conclude that the proportion of flights that are over 30 minutes late is different between flying United or Delta. (a) State the null and alternative hypothesis. (b) What statistic will be recorded for each of the simulated samples to create the randomization distribution? What is the value of that statistic for the observed sample? (c) Use StatKey or other technology to create a randomization distribution. Estimate the p-value for the observed statistic found in part (b). (d) At a significance level of \(\alpha=0.01\), what is the conclusion of the test? Interpret in context. (e) Now assume we had only collected samples of size \(75,\) but got essentially the same proportions (5/75 late flights for Delta and \(12 / 75\) late flights for United). Repeating steps (b) through (d) on these smaller samples, do you come to the same conclusion?
In Exercise 3.89 on page \(239,\) we found a \(95 \%\) confidence interval for the difference in proportion of rats showing compassion, using the proportion of female rats minus the proportion of male rats, to be 0.104 to \(0.480 .\) In testing whether there is a difference in these two proportions: (a) What are the null and alternative hypotheses? (b) Using the confidence interval, what is the conclusion of the test? Include an indication of the significance level. (c) Based on this study would you say that female rats or male rats are more likely to show compassion (or are the results inconclusive)?
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