Graph the probability density function of a binomial distribution with n = 100 and p = 1/6. Superimpose a graph of the approximating normal probability density function. Use vertical bars to show the binomial probabilities and use a smooth line in a different colour for the normal curve. Compute the exact binomial probability of getting 10 or less. Indicate this tail of the distribution by using a different colour for the vertical bars. Compare the exact calculation with the normal approximation, computed with and without the continuity correction. Repeat with n = 10, this time computing the probability of getting 1 or less.
Generate 3 samples of n = 10 pseudorandom observations from a normal distribution with mean 100 and standard deviation 10. Test for normality graphically by plotting, for each of the three samples, a histogram with the fitted normal density superimposed and a probability plot with fitted line. Repeat for n = 20, 40, 100, 1000. How many observations do you need in order to say with any confidence that the data came from a normal distribution?
Repeat Question 2 but this time generate the data from an exponential distribution with mean 100. How many observations do you need in order to say with any confidence that the data did not come from a normal distribution?
Demonstrate the Central Limit Theorem by generating 500 samples, each of size n = 100, from an exponential distribution with mean 100. Compute the mean of each sample. Display the 500 sample means on a histogram and on a normal probability plot. Find the mean and standard deviation of this distribution. Are your results consistent with the Central Limit Theorem?
The weight of a tablet you are producing follows a normal distribution with mean 100 g and standard deviation 3 g. Only tablets between 94 g and 106 g are acceptable. In a box of 50 tablets, what is the probability that more than 3 will be unacceptable? What would this probability become if the standard deviation were reduced to 2 g?
3-80 (p 90)
4-106 (p 142)
5-54 (p 173) - use qqnorm() and qqline() as probability paper is no longer easily found!
5-74 (p 185)
5-132 (p 198)