Statistics 371 Brief Solutions #5 Fall 2002
1. Graph the binomial distribution for n= 10 and p= 0.1 with the command gbinom(10,0.1). Repeat this for p=
0.2,0.3, . . . , 0.9. [Please do not include these graphs with your assignment.]
(a) How does the center of the distribution change as pchanges?
Solution: The center moves to the right as pincreases. (The mean is 10p.)
(b) For which value of pis the distribution most strongly skewed right? left? most symmetric?
Solution: The distribution has the strongest skew right when pis small, 0.1 for this problem. The distribution has the
strongest skew left when pis large, 0.9 for this problem. The distribution is perfectly symmetric when p= 0.5.
(c) For which value of pis the standard deviation the largest?
Solution: The standard deviation is largest when the graph is most spread out around the mean. This happens when
p= 0.5. (You could use the formula to find σ=pnp(1 −p)if you wanted to verify numerically what your eye tells
you.)
2. Graph the binomial distribution for n= 1 and p= 0.5 with the command gbinom(1,0.5). Repeat this for n=
2,4,8,16,32,64,128. [Please do not include these graphs with your assignment.]
(a) How does the center of the distribution change as nchanges?
Solution: The center increases as nincreases. (The mean is np.)
(b) Is this distribution skewed for any n?
Solution: With p= 0.5, the distribution is perfectly symmetric for all n.
(c) What is the smallest nfor which the distribution looks approximately normal? (There is no single correct answer.)
Solution: To my eye, when n= 16 I can see the bell shape appear, although it is somewhat apparent for smaller nand
even more apparent for larger n. Many different answers could be correct.
(d) What happens to the range of values for which the probabilities are large enough to be visible as nincreases?
Solution: The range of visible probabilities (the number of visible lines) increases as nincreases.
(e) What happens to the range of values for which the probabilities are large enough to be visible over nas nincreases?
Solution: The range of visible probabilities occupies a smaller and smaller portion of the entire possible range from 0 to
nas nincreases.
3. Graph the binomial distribution for n= 1 and p= 0.1 with the command gbinom(1,0.1). Repeat this for n=
2,4,8,16,32,64,128. [Please do not include these graphs with your assignment.] About how large does nneed to be
before the distribution looks nearly symmetric and approximately normal? Compare your answer here to the answer
in part (c) in the previous problem.
Solution: With n= 16, there is still a noticeable moderately strong skew. By the time n= 64, the skew has mostly
disappeared and the visible probabilities look to be fairly symmetric.
4. Exercise 5.3 (page 157).
Solution: In a forest, 25% of the white pines have blister rust. Four white pines are sampled and ˆpis the sample proportion
with blister rust. Find and plot the probability distribution of ˆp.
> prob <- dbinom(0:4, 4, 0.25)
> prob
[1] 0.31640625 0.42187500 0.21093750 0.04687500 0.00390625
> plot((0:4)/4, prob, type = "h")
Bret Larget October 7, 2002
