Question

Which of the following are true and explain your answer using the Normal Curve?

Hint: Properties of the Normal Curve and Empirical Rule. Note: The Normal curve has two tales. For example, ±1 standard deviation around the mean means that there is 34.13% of the curve on each side of the normalized 0. Or the area (probability) under the curve = 68.26%

A)19 of every 20 observations would fall between ±2 standard deviations around the mean.

B) 2 of every 3 observations would fall between ±1 standard deviation around the mean.

C) 4 of every 5 observations would fall between ±1.28 standard deviations around the mean.

Answer 1

Empirical Rule:

68% of data falls within the first standard deviation from the mean.

95% fall within two standard deviations.

99.7% fall within three standard deviations.

A) 19 out of 20 observation i.e 19/20 = 0.95 i.e 95% of observations ; Under the empirical rule ; 95% fall within two standard deviations of the mean

i.e

19 out of 20 observation would fall between ±2 standard deviations around the mean : True

B) 2 of every 3 observations i.e 2/3= 0.67 i.e 67% of observations ;

Under the empirical rule ; 68.26% fall within one standard deviations of the mean would fall between ±1 standard deviation around the mean.

No. As 67% is not equal to 68.26% ;

If 67% is considered approximately equal to 68.26% then yes.

C)4 of every 5 observations i.e 4/5 = 0.8

Let us find the area under the  ±1.28 standard deviations around the mean

i.e

P(-1.28 < Z < 1.28) = P(Z<1.28) - P(Z<-1.28)

From standard normal tables ,

P(Z<1.28) = 0.8997

P(Z<-1.28) = 0.1003

P(-1.28 < Z < 1.28) = P(Z<1.28) - P(Z<-1.28) = 0.8997 - 0.1003 = 0.7994 0.80

4 of every 5 observations would fall between ±1.28 standard deviations around the mean : True