Question

1.90 Gross domestic product. Refer to Exercise
1.46, where we examined the gross domestic product
of 189 countries. GDP
(a) Compute the mean and the standard deviation.
(b) Apply the 68–95–99.7 rule to this distribution.
(c) Compare the results of the rule with the actual
percents within one, two, and three standard deviations
of the mean.
(d) Summarize your conclusions.

mean is 380,773 and standard deviation is 1450933

Answer 1

Solution-b:

sample mean=xbar=mu=380773

sample sd=s= 1450933

According to 68-95-99.7% rule

68% of the values lies within one standard deviation of the mean

=mean-sd and mean+sd

380773-1450933 and 380773+1450933

-1070160 and  1831706

95% of the values lies within twostandard deviation of the mean

=mean-2sd and mean+2sd

-2521093 and  3282639

99.7% of the values lies within three standard deviation of the mean

=mean-3sd and mean+3sd

-3972026 and  4733572

Soluion-c:

mean is 380773 and standard deviation is 1450933

P(-1070160<X<1831706) to get one sd of the mean probability

P(-2521093<X<3282639) to get two sd of the mean probability

P(-3972026 <X<4733572) to get three sd of the mean probability

Rcode to get the probbaility

library(tigerstats)
pnormGC(bound=c(-1070160,1831706),region="between", mean=380773,sd=1450933,graph=TRUE)
pnormGC(bound=c(-2521093, 3282639),region="between", mean=380773,sd=1450933,graph=TRUE)
pnormGC(bound=c( -3972026, 4733572),region="between", mean=380773,sd=1450933,graph=TRUE)

Output:

actual percents within one standard deviations of the mean= 0.6826895*100=68.3%

actual percents within two standard deviations of the mean= 0.9544997*100=95.4%

actual percents within three standard deviations of the mean= 0.9973002*100=99.7%

the results of the rule with the actual
percents within one, two, and three standard deviations
of the mean are same