Question

SupposeZ∼N(0, 1). Use R to find:(a) What percent of the area under the density curve is within±one standard devi-ation of the mean (i.e.,P(μ−σ < Z < μ+σ))?(b) What percent of the area under the density curve is within±2σof the mean?(c) What percent of the area under the density curve is within±3σof the mean?(d) Are these percentages the same forX∼N(μ=−15,σ2= 50)?

Answer 1

Following is the screen shot of the R-script

# Creating mean and standard deviation variable #(a) per(pnorm(m+s,mean-m,sd-s)-pnorm(m-s,mean-m,sd-s)) 100 #requried percenatge is round (per,2) # (b) per = (pnor m(m-2°s , mean-m ,sd-s)-pnor m(m-2°s,mean-m ,sd-s))"100 #requried percenatge is round (per,2) #(c) per(pnorm(m+3 s, mean-m,sd-s) -pnorm(m-3*s, mean-m,sd-s)) #requried percenatge is 100 round (per,2) # Changing values of mean and standard deviation variable m15 s-sqrt (50) per(pnorm(m+s,mean-m,sd-s)-pnorm(m-s,mean-m,sd-s)) 100 #requried percenatge is round (per,2) per(pnorm(m+2*s, mean-m,sd-s) -pnorm(m-2*s, mean-m,sd-s)) 100 #requried percenatge is round (per,2) per(pnorm(m+3 s, mean-m,sd-s) -pnorm(m-3*s, mean-m,sd-s)) 100 #requried percenatge is round (per,2)

Following is the R-output:

> # creating mean and standard deviation variable >s=1 > #(a) > per- (pnorm(m+s, mean-m,sd-s)-pnorm(m-s,mean-m,sd-s))*100 > #requried percenatge is > round (per,2) [1] 68.27 > #(b) > per (pnor m(m+ 2 * s , mean-m , sd-s)-pnor m(m_2*s , mean=m , sd=s))*100 > #requried percenatge is > round (per,2) [1] 95.45 > #(c) > per- (pnorm(m+3*s, mean-m,sd-s)-pnorm(m-3*s,mean-m,sd-s))*100 > #requried percenatge is > round (per,2) [1] 99.73 > #(d) > # Changing values of mean and standard deviation variable >m=-15 s-sqrt (50) > per- (pnorm(m+s, mean-m,sd-s)-pnorm(m-s,mean-m,sd-s))*100 > #requried percenatge is > round (per,2) [1] 68.27 > per- (pnorm(m+2*s,mean-m,sd-s)-pnorm (m-2 s,mean-m,sd-s))*100 > #requried percenatge is round (per,2) [1] 95.45 > per- (pnorm(m+3*s, mean-m,sd-s)-pnorm(m-3*s,mean-m,sd-s))*100 > #requried percenatge is > round (per,2) [1] 99.73

Chaining mean and SD did not change the percentages.