How important is the assumption "The sampled population is normally distributed" to the use of Student's t-distribution? Using a computer, simulate drawing 100 samples of size 10 from each of three different types of population distributions, namely, a normal, a uniform, and an exponential. First generate 1000 data values from the population and construct a histogram to see what the population looks like. Then generate 100 samples of size 10 from the same population; each row represents a sample. Calculate the mean and standard deviation for each of the 100 samples. Calculate t_ for each of the 100 samples. Construct histograms of the 100 sample means and the 100 t_ values. (Additional details can be found in the Student Solutions Manual.) For the samples from the normal population:
a. Does the distribution appear to be normal? Find percentages for intervals and compare them with the normal distribution.
b. Does the distribution of t_ appear to have a t-distribution with df = 9? Find percentages for intervals and compare them with the t-distribution. For the samples from the rectangular or uniform population:
c. Does the distribution appear to be normal? Find percentages for intervals and compare them with the normal distribution.
d. Does the distribution of t_ appear to have a t-distribution with df = 9? Find percentages for intervals and compare them with the t-distribution. For the samples from the skewed (exponential) population:
e. Does the distribution appear to be normal? Find percentages for intervals and compare them with the normal distribution.
f. Does the distribution of t_ appear to have a t-distribution with df = 9? Find percentages for intervals and compare them with the t-distribution. In summary:
g. In each of the preceding three situations, the sampling distribution for appears to be slightly different from the distribution of t_. Explain why.
h. Does the normality condition appear to be necessary in order for the calculated test statistic t_ to have a Student's t-distribution? Explain