MIT OpenCourseWare http://ocw.mit.edu 18.443 Statistics for Applications Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.18.443 problems based on Chapter 6 and Section 8.5.3 of Rice 1. (a) Rice, p. 198, Problem 3, but with 49 instead of 16 and 0.8 instead of 0.5. More explicitly: Let X be the average of a sample of 49 independent normal random variables with mean 0 and variance 1. Determine c such that P (|X| < c) = 0.8. (b) Rice, p. 198, Problem 4, but for t4 instead of t7. More explicitly: If T follows a t4 distribution, find t0 such that (a) P (|T | < t0) = .9 and (b) P (T > t0) = 0.05. 2. Small measurement errors are believed to follow a normal distribution. The follmeasurements of the constant were published, one in 1981, and the rest in 1998-9, inof 10−11N·m2/kg2 (N=Newtons, m=meters, kg=kilograms). 6.1 ± 0.4, 1981 England, Page and Geilker 6.6729 ± 0.0005, 1998 Russia, Karagioz et al. 6.6735 ± 0.0029, 1998 Germany, Kleinevoß et a l. 6.6873 ± 0.0094, 1998 USA, Schwarz et al. 6.6699 ± 0.0007, 1999 China, Luo et al. 6.6742 ± 0.0006, 1999 New Zealand, Fitzgerald & Armstrong 6.6830 ± 0.0114, 1999 England, Richman et al. 6.6754 ± 0.0015, 1999 Switzerland, N olting et al. owing units In this problem, ignore the error bars such as ±0.0007 given by the experimenters and just consider the numbers in the leftmost column as data points. According to the results from the Shapiro-Wilk test handed out, the 8 observations are not normally distributed. Just consider the 7 from 1998-99, for which normality was not rejected. (a) Find the sample mean X of these 7 observations. (b) Fi nd their sample variance S2 ≡ s2 X and standard deviation sX . Hint: scientific calculators often give s = sX ; square it to get sX 2 . (c) Find (X − 6.1)/sX to see, in terms of sample standard deviations for the 7 obser-vations, how far 6.1 is away from their mean. (This indicates how much of an outlier it is.) (d) Find a 95% confidence int erval for the gravitational constant (the true mean of the observations), based on the 7 data points from 1998-99, assuming a normal distribution. 3. For this problem, we’ll study the errors, in other words the variances. Again omit the 1981 Page and Geilker numbers. For the 7 o bservations from 1998-1999, assuming that they are i.i.d. normal with the same but unknown mean µ and variance σ2 , (a) Give a 95% confidence interval for σ2 . Take the square roots of the endpoints t o get a 95% confidence int erval for σ. (b) In this part only, consider the estimated experimental errors for the seven obser-vations, given after ±. Which of them are within the confidence interval for σ? 4. In an experiment, 26 women in good health had temperatures (taken by an oral ther-mometer) withX = 98.4 1◦ F. and sample standard deviation sX = 0.77◦ F. For 122 men, the sample mean was Y = 98.10◦ F., a nd sY = 0.72◦ F. Assume that the temperature distributions for men, and for women, are normal, although they might be different, with different means and/or variances for the two genders. From this information give 95% confidence intervals for (a) µX , women’s average temperature, and (b) µY , men’s average temperature. (c) Do these intervals overlap? Does eit her cont ain 9 8.6◦ F.? 5. Submit your answer t o this problem as a printout on a separate page, with your name on it, not fastened to the rest of your problem set. In problem 2 and the related handout, we saw that a data set with as few as 9 points can be clearly non-normal because of an outlier. But there are many other ways a distribution can depart from being normal. In fact if the observations are i.i.d. with any distribution other than a norma l one, the Shapiro-Wilk test can detect that for a large enough sample. Get into R (in Athena, by “add r” then “R”, without quotes). (Also, by the way, R is freely available public software, available on websites called CRAN.) When you’re ready to generate your file for printed o utput, give the command sink(”myfile”) or whatever you want to call your file, then when you’re done type sink() and then q() to get out of R. The file will have ma terialized in whatever directory you’re in and you can print it. The “>” at the beginning of each line with R code is a system prompt which could be different in different systems. You may want to try the foll owing commands in R and see results on the screen to find how things work, before you produce your file to turn in. (a) generate some i.i.d. standard exponential variables as follows. (Recall that a standard exponential distribution has density e−x for x ≥ 0 and 0 for x < 0.) > x <- rex p(25) Then x will be a data vector consisting of 25 i.i.d. standard exponential variables. Here “exp” tell s the system to use the exponential distribution, the “r” before it says to generate random variables i. i.d. with that distribution, and the 25 (which could be replaced by any positive integer) g ives how many to generate. Apply the Shapiro-Wilk test: > shapiro.test(x) The deviation o f a data set from normality is by default “significant” if the p-value is ≤ 0.05, or “highly significant” if it’s ≤ 0. 001. What do you find? The exponential distribution is very different from a normal in that it’s an asymmetric, skewed distribution with a sharp cutoff on the left at 0 but extending arbitrarily far out to the right. Its mean is 1, but its median, having equal probability 1/2 to the left or right of it, is ln 2 < 1. (b) Next let’s try the same thing but for the uniform U[0, 1] distribution. This has density f(x) = 1 for 0 ≤ x ≤ 1 and 0 elsewhere. Unlike the exponential, this distribution is symmetric around its mean, which