Section 15 Solutions

15. The Normal Distribution

Question 1

From Section 9.5 we know the height of the uniform density is 1/40. The probability to the right of a particular value x is

P(X ≥ x) = base × height = (190 - x) × (1/40).

From this we can build the following probability table:

	Digits of x
x	0	1	2	3	4	5	6	7	8	9
150	1.000	.975	.950	.925	.900	.875	.850	.825	.800	.775
160	.750	.725	.700	.675	.650	.625	.600	.575	.550	.525
170	.500	.475	.450	.425	.400	.375	.350	.325	.300	.275
180	.250	.225	.200	.175	.150	.125	.100	.075	.050	.025

Question 2

From Section 9.5 we know the height of the triangular density is 1/20. If x is 170 or more then the height of the density curve at x is (1/20)×((190 - x)/20). You can check that this gives height 0 for x=190 and height 1/20 for x=170. The area to the right of x is then

P(X ≥ x) = (1/2) × base × height = (1/2) × (190 - x) × (1/20)×((190 - x)/20) = (1/800)×(190 - x)².

Similarly, for x ≤ 170 the area to the right is

P(X ≥ x) = 1 - (1/800)×(x - 150)².

From these we can build the following probability table:

	Digits of x
x	0	1	2	3	4	5	6	7	8	9
150	1.000	.999	.995	.989	.980	.969	.955	.939	.920	.899
160	.875	.849	.820	.789	.755	.719	.680	.639	.595	.549
170	.500	.451	.405	.361	.320	.281	.245	.211	.180	.151
180	.125	.101	.080	.061	.045	.031	.020	.011	.005	.001

Question 3

Unlike the previous two questions, for Normal probabilities it is not possible to write down a formula for the area to the right of a particular x. Instead we will standardize our x values, using μ = 167 and σ = 6.6, and then use the table of standard Normal probabilities (Table 15.1).

For example, to find P(X ≥ 170) we calculate z = (170 - 167)/6.6 = 0.45. Table 15.1 then gives P(Z ≥ 0.45) = .326.

	Digits of x
x	0	1	2	3	4	5	6	7	8	9
150	0.995	0.992	0.988	0.983	0.976	0.966	0.953	0.936	0.913	0.887
160	0.855	0.819	0.776	0.729	0.674	0.618	0.560	0.500	0.440	0.382
170	0.326	0.271	0.224	0.181	0.145	0.113	0.087	0.064	0.047	0.034
180	0.024	0.017	0.012	0.008	0.005	0.003	0.002	0.001	0.001	0.000

Question 4

Four boxplots are shown below. There is certainly a lot of variability in the kind of picture you get from a boxplot when the sample size is so small.