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4.11 Exercises using barcode drawer for none control to generate, create none image in none applications. Intelligent Mail Maindonald, J. none for none H. 1992.

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Essentials of Statistical Inference. 4.11 Exercises.

1. Using the da ta set nswdemo (DAAG), determine 95% con dence intervals for: (a) the 1975 mean incomes of each group; (b) the 1978 mean incomes of each group. Finally, calculate a 95% con dence interval for the difference in mean income between treated and controls in 1978.

2. Draw graphs that show, for degrees of freedom between 1 and 100, the change in the 5% critical value of the t-statistic. Compare a graph on which neither axis is transformed with a graph on which the respective axis scales are proportional to log(t-statistic) and log(degrees of freedom).

Which graph gives the more useful visual indication of the change in the 5% critical value of the t-statistic with increasing degrees of freedom 3. Generate a random sample of 10 numbers from a normal distribution with mean 0 and standard deviation 2. Use t.

test() to test the null hypothesis that the mean is 0. Now generate a random sample of 10 numbers from a normal distribution with mean 1.5 and standard deviation 2.

Again use t.test() to test the null hypothesis that the mean is 0. Finally write a function that generates a random sample of n numbers from a normal distribution with mean and standard deviation 1, and returns the p-value for the test that the mean is 0.

4. Use the function that was created in Exercise 3 to generate 50 independent p-values, all with a sample size n = 10 and with mean = 0. Use qqplot(), with the argument setting x = qunif(ppoints(50)), to compare the distribution of the p-values with that of a uniform random variable, on the interval [0, 1].

Comment on the plot. 5. The following code draws, in a 2 2 layout, 10 boxplots of random samples of 1000 from a normal distribution, 10 boxplots of random samples of 1000 from a t-distribution with 7 d.

f., 10 boxplots of random samples of 200 from a normal distribution, and 10 boxplots of random samples of 200 from a t-distribution with 7 d.f.

:. oldpar <- pa none for none r(mfrow=c(2,2)) tenfold1000 <- rep(1:10, rep(1000,10)) boxplot(split(rnorm(1000*10), tenfold1000), ylab="normal - 1000") boxplot(split(rt(1000*10, 7), tenfold1000), ylab=expression(t[7]*" - 1000")) tenfold100 <- rep(1:10, rep(100, 10)) boxplot(split(rnorm(100*10), tenfold100), ylab="normal - 100") boxplot(split(rt(100*10, 7), tenfold100), ylab=expression(t[7]*" - 100")) par(oldpar). A review of inf erence concepts Refer back to the discussion of heavy-tailed distributions in Subsection 3.2.2, and comment on the different numbers and con gurations of points that are agged as possible outliers.

. 6. Here we gene none none rate random normal numbers with a sequential dependence structure: y1 <- rnorm(51) y <- y1[-1] + y1[-51] acf(y1) # acf is autocorrelation function # (see 9) acf(y) Repeat this several times. There should be no consistent pattern in the acf plot for different random samples y1.

There will be a fairly consistent pattern in the acf plot for y, a result of the correlation that is introduced by adding to each value the next value in the sequence. 7. Create a function that does the calculations in the rst two lines of the previous exercise.

Put the calculation in a loop that repeats 25 times. Calculate the mean and variance for each vector y that is returned. Store the 25 means in the vector av, and store the 25 variances in the vector v.

Calculate the variance of av. 8. The following use the data frame nswpsid3, created as in footnote 8: (a) For each column of the data set nswpsid3 after the rst, compare the control group (trt==0) with the treatment group (trt==1).

Use overlaid density plots to compare the continuous variables, and two-way tables to compare the binary (0/1) variables. Where are the greatest differences (b) Repeat the comparison, but now for the data set nswdemo. (c) Compare and contrast the two sets of results.

Read carefully the help pages for psid3 and for nswdemo, and comment on why the different thrust of the two sets of results is perhaps not surprising. 9. In a study that examined the use of acupuncture to treat migraine headaches, consenting patients on a waiting list for treatment for migraine were randomly assigned in a 2:1:1 ratio to acupuncture treatment, a sham acupuncture treatment in which needles were inserted at non-acupuncture points, and waiting-list patients whose only treatment was self-administered (Linde et al.

, 2005). (The sham acupuncture treatment was described to trial participants as an acupuncture treatment that did not follow the principles of Chinese medicine.) Analyze the following two tables.

What, in each case, are the conclusions that should be drawn from the analyses Comment on implications for patient treatment and for further research: (a) Outcome is classi ed according to numbers of patients who experienced a greater than 50% reduction in headaches over a four-week period, relative to a pre-randomization baseline:.
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