Homework 7

load(url("http://www.rossmanchance.com/iscam3/ISCAM.RData"))
library(dplyr)
library(ggplot2)
homework at: http://www.rossmanchance.com/iscam3/instructors.html

Chapter 3 HW 8 Praising Intelligence or Effort

Psychologists investigated whether praising a child’s intelligence, rather than praising his/her effort, tends to negative consequences such as undermining their motivation (Mueller & Dweck, 1998). Children participating in the study were given a set of problems to solve. After the first set of problems, half of the children were randomly assigned to be praised for their intelligence, whereas the other half was praised for their effort. The children were then given another set of problems to solve and later told how many they got right. They were then asked to write a report about the problems for other children to read, including information about how many they got right. Some of the children misrepresented (i.e., lied about) how many they got right, as shown in the following table:

  1. Identify the explanatory and response variables in this study.
  2. For each group, determine the proportion who lied, and identify them with appropriate symbols.

  3. Describe how you could use index cards to conduct a simulation analysis for determining whether the difference between these proportions is statistically significant. Include the following information in your description:

    1. how many cards you would use
    2. how many would be marked how
    3. how many you would deal out
    4. which kinds of cards you would count
    5. what you would compare the results to, after you conducted a large number of repetitions

  4. Use the Analyzing Two-way Tables applet to conduct a simulation with 1000 repetitions. Submit a screen capture of the resulting histogram, and report the empirical p-value from the applet.

  5. Use R to perform Fisher’s Exact Test. Along with reporting the p-value, provide an appropriate graph, and also express the p-value as P(X ____ k), where you:
    1. insert the appropriate inequality in the _____ space,
    2. report the appropriate value of k,
    3. indicate what kind of probability distribution X has (name),
    4. provide the numerical values (inputs) associated with that probability distribution.
  6. Provide a complete, detailed interpretation (in one or two sentences) of what this p-value means in this context (i.e., what is it the probability of, assuming what?)
  7. Based on this p-value, is the observed difference between the groups statistically significant at the α = 0.05 level? Explain how you know.

  8. Summarize and justify your conclusion about whether the data provide evidence that praising a child’s intelligence leads to more negative consequences than praising his/her effort.

Chapter 3 HW 7 Effectiveness of AZT

In 1993, one of the first studies aimed at preventing maternal transmission of AIDS to infants gave the drug AZT to pregnant, HIV-infected women (Connor et al., 1994). Roughly half of the women were randomly assigned to receive the drug AZT, and the others received a placebo (a “fake” treatment, same appearance as the drug but with no active ingredients). The HIV-infection status was then determined for 363 babies, 180 from the AZT group and 183 from the placebo group. Of the 180 babies whose mothers had received AZT, 13 were HIV-infected, compared to 40 of the 183 babies in the placebo group.

  1. Identify the observational units in this study.
  2. Explain why you think use of a placebo would be important in a study like this.
  3. If you were going to carry out a simulation to analyze these data, how many player cards would you have? How many would be marked “red” and how many “blue”? How many would you deal out? What “statistic” would you record for each shuffle? How would you determine the p-value?
  4. Suppose instead of measuring “whether or not the baby is HIV-infected,” the researchers gave a numerical score to the health of each baby. You want to know whether the difference in the mean scores given to these two groups is statistically significant. Explain briefly how you would modify your simulation.

Chapter 3 HW 9 The Winner?

Enamored of the solitaire game on his new computer, author A sets out to estimate his probability of winning the game and wins 25 games while losing 192 games. Anxious to outperform author A, author B plays 444 games of solitaire and wins 74.

  1. Do these data arise from sampling from two random processes or random sampling from two populations?
  2. Set up the null and alternative hypotheses for comparing the success probabilities of these authors.
  3. Calculate and interpret a 95% confidence interval for the difference in success probabilities between these two authors (no Wilson adjustment from the end of Investigation 3.1). Make sure it’s clear how the interval was calculated and include your output.
  4. Based on your confidence interval, we will reject or fail to reject the null hypothesis in (b) at the 5% level of significance?
  5. In particular, are you convinced author B is a better solitaire player than author A? Explain.