Probability and Statistics for Computer Scienc

Question 1 The following are the percentages of ash content in 12 samples of coal found in close proximity:

9.1, 16.5, 12.6, 16.8, 21.3, 9.6, 14.1, 22.8, 14.3, 20.4, 12.1, 18.9
Find the
a) sample mean,
b) sample median, and
c) sample standard deviation of these percentages.

Question 2  The following data represent the lifetimes (in hours) of a sample of 40 transistors:
112, 121, 126, 108, 141, 90, 136, 134, 121, 118,
108, 143, 108, 122, 127, 140, 113, 117, 126, 130,
134, 120, 131, 133, 118, 125, 151, 147, 137, 140,
132, 119, 110, 124, 132, 162, 135, 130, 136, 128
a) Determine the sample mean, median, and mode.
b) Determine the 1st and 3rd quartiles, the interquartile range, and any outliers.
c) Give a cumulative relative frequency plot of these data.
d) Treating the cumulative relative frequency plot as the Cumulative Probability
Distribution of an unknown continuous random variable, estimate the median of this random variable from the plot.

Question 3  The average particulate concentration, in micrograms per cubic meter, was measured in a petrochemical complex at 36 randomly chosen times, with the following concentrations resulting:
5, 18, 15, 77, 133, 220, 130, 85, 103, 125, 80, 107, 124, 106, 113, 165, 137, 125,
124, 65, 82, 95, 77, 115, 70, 110, 144, 128, 133, 81, 129, 114, 45, 92, 117, 153
a) Represent the data in a histogram.
b) Is the histogram approximately normal?
c) Calculate the sample mean
.
d) Calculate the sample standard deviation S.
e) Determine the proportion of the data values that lies within
± 1.5 and compare with the lower bound given by Chebyshev's inequality.
f) Determine the proportion of the data values that lies within
± 2 and compare with the lower bound given by Chebyshev's inequality.

Question 4 An instructor knows from past experience that student exam scores have mean 77 and standard deviation 15. At present the instructor is teaching two separate classes - one of size 25 and the other of size 64. 
a) Approximate the probability that the average test score in the class of size 25 lies between 72 and 82.
b) Repeat part (a) for a class of size 64.
c) What is the approximate probability that the average test score in the class of size 25 is higher than that of the class of size 64?
d) Suppose the average scores in the two classes are 76 and 83. Which class, the one of size 25 or the one of size 64, do you think was more likely to have averaged 83?

Question 5 Each computer chip made in a certain plant will, independently, be defective with probability .25. If a sample of 1,000 chips is tested, what is the approximate probability that fewer than 200 chips will be defective?

Question 6 The following are scores on IQ tests of a random sample of 18 students at a large eastern university.
130, 122, 119, 142, 136, 127, 120, 152, 141,
132, 127, 118, 150, 141, 133, 137, 129, 142
a) Construct a 95 percent confidence interval estimate of the average IQ score of all students at the university.
b) Construct a 95 percent lower confidence interval estimate.
c) Construct a 95 percent upper confidence interval estimate.
Question 7 [12 marks] Each of 20 science students independently measured the melting point of lead. The sample mean of these measurements was 330.2 degrees centigrade.
a) If the standard deviation of such measurements is known to be 14, find a 95 percent twosided confidence interval estimate of the true melting point of lead.
b) Suppose that the population variance is not known in advance. If the sample standard deviation is 15.4 degrees centigrade, compute a 95 percent two-sided confidence interval of the true melting point of lead.
Question 8 [12 marks] The capacities (in ampere-hours) of 10 batteries were recorded as follows:
140, 136, 150, 144, 148, 152, 138, 141, 143, 151
a) Estimate the population variance a² .
b) Compute a 99 percent two-sided confidence interval for a²  .
c) Compute a value v that enables us to state, with 90 percent confidence, that a²  is less than v.