invest_3ed.pdf

# Chancerossman 2015 iscam iii vs book on tape so we

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Chance/Rossman, 2015 ISCAM III Example 4.3 315 vs. book-on-tape, so we have extremely strong evidence that the cell phone really does increase reaction times. (d) Because the sample size (16) is fairly small, the t -procedures are valid only if the population of differences follows a normal distribution. The dotplot of differences from these 16 subjects looks roughly symmetric, so the t -procedures are probably valid to apply here. (e) The test statistic is d d d n s x 0 ± = 16 331 . 51 125 . 47 ≈ 3.67. The p -value is the probability that a t - distribution with 15 degrees of freedom is 3.67 or larger; R reveals this p-value to be 0.001137. This p- value is very small, so we would reject the null hypothesis at the 0.05 and 0.01 significance levels. The experimental data provide very strong evidence that talking on a cell phone does cause an increase in mean reaction time, as compared to listening to a book-on-tape. The cause/effect conclusion is justified because this is a randomized experiment with a very small p-value. Note: This evidence is stronger than what we saw with the sign test in Example 2.2. (f) A 95% confidence interval for the population mean difference P d is: d x ± t * × d d n s , which is 47.125 ± 2.131(51.331)/ 16 , which is 47.125 ± 27.347, which is (19.778, 74.472). We can be 95% confident that the mean reaction time while talking on a cell phone is between roughly 20 and 75 milliseconds longer than when listening to a book-on-tape. A 95% prediction interval for the difference in reaction times for a particular subject is: d x ± t * d d n s / 1 1 ² , which is 47.125 ± 2.131(51.331) 16 / 1 1 ² , which is 47.125 ± 112.753, which is ( ± 64.628, 159.878). We can be 95% confident that the an individual subject will react anywhere from 65 milliseconds more quickly to 160 milliseconds more slowly talking on a cell phone as compared to listening to a book-on-tape. (g) An advantage of the sign test from Chapter 2 is we don’t need to rely on large sample sizes or normality of the differences for the procedure to be valid. However, by ignoring some information in the data, we would expect this procedure to have lower power and we lose information about the size of the difference. The conditions for the t- procedure seem to be satisfactorily met here.

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Chance/Rossman, 2015 ISCAM III Chapter 4 Summary 316 CHAPTER 4 SUMMARY In this chapter, you have again focused on comparing the outcomes for different groups, but for quantitative data. The tools for descriptive statistics are the same as in chapter 2 (e.g., means, medians, standard deviations, interquartile ranges, dotplots, histograms, and boxplots). After exploring and comparing the groups descriptively, it may be appropriate to ask whether the differences observed could have arisen “by chance alone .” In other words, is the observed difference large enough to convince us that it arose from a genuine difference in the groups instead of from the randomness inherent in designing the experiment or in selecting the samples? Analogous to comparing proportions in Chapter 3, we used simulation to approximate how often we would obtain a difference in means at least as extreme as observed just by chance (random sampling or random assignment). Then
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