STAT 705 Spring 2017
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Homework 7
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The Microcomputer Data set is on the course web page.

Do Problems 12.9 and 12.10 in the textbook, and ALSO do the following problems:

1.  Consider an sample (of size n) of i.i.d. Binomial(1, pi) observations.  
We showed in class that the sample proportion of "successes" is the least-squares 
estimator of pi.  Show that it is also the maximum likelihood estimator of pi.
(Hint:  It is easier to maximize the logarithm of the likelihood function.)

2.  A traffic study was done to chart accidents in the Rosewood neighborhood.
For a random sample of 110 days (taken over 3 years) the following data were 
observed:  27 days had zero accidents; 38 days had one accident; 23 days had 
two accidents; 14 days had three accidents; 8 days had four or more accidents.

(a) Johnson believed that for the population of interest:  one-fourth of days 
had no accidents; one-third of days had one accident; one-fifth of days 
had two accidents; one-tenth of days had three accidents; and the remainder of 
days had four or more.  State the hypotheses for testing Johnson's claim.  Also 
perform the test, stating the test statistic value, P-value, and your conclusion.

(b) Smith believed that a Poisson model with mean lambda=1.5 would be an excellent
model for these data.  Use the appropriate Poisson probabilities to determine 
the null hypothesis that reflects Smith's belief.  Perform the test, stating the 
test statistic value, P-value, and your conclusion.

(c) Find a 95% CI for the probability that a random day has no accidents.  
Identify the specific CI method that you use.

3.  A similar study was done over the same 3-year period in the Summit 
neighborhood.  For a random sample of 153 days, the following data were 
observed:  54 days had zero accidents; 56 days had one accident; 23 days had 
two accidents; 12 days had three accidents; 8 days had four or more accidents.

(a) It was believed that the true proportion of days with no accidents was greater 
in the Summit neighborhood than in the Rosewood neighborhood.  To check this, 
perform the appropriate hypothesis test, using alpha = 0.05 and stating the 
test statistic value, P-value, and your conclusion.

(b)  For the Summit data, each sampled day was additionally classified as either 
rainy or sunny.  The following two-way table was obtained.

            Number of Accidents
        0       1      2      3      4+
      -----------------------------------
Rain |  16      15     10     5      4   |
Sun  |  38      41     13     7      4   |
      -----------------------------------


Is there significant evidence that the number of daily accidents in the Summit 
neighborhood depends on the weather?  Perform the appropriate Chi-squared test, using 
alpha = 0.05 and stating the test statistic value, P-value, and your conclusion.

(c)  Are the assumptions for the Chi-squared test met with this data set?  Perform 
the corresponding small-sample test; are the conclusions from this test different?

4.  Greg believed that certain observations on peas could be classified into three 
categories, with respective cell probabilities 0.60, 0.20, and 0.20.  Greg observed 
1000 peas and reported that of these, 602 observations fell in the first category, 
197 fell in the second category, and 201 fell in the third category.  Ronnie accused 
Greg of falsifying his data so that they would better match his prior belief.
Suggest a way that Ronnie could statistically demonstrate that Greg "cooked" the
pea data.  For Greg's data set, what would be Ronnie's final conclusion?