Question

Question 11 For any normally distributed data: (Write the answers without % sign) A: About what percent of data is within 1 standard deviation of the mean? B: About what percent of data is within the mean and 2 standard deviations above the mean? C: About what percent of data is more than 2 standard deviations above the mean?

Answer 1

Dear student, We can provide you with the solution of 4 sub question at a time.

11) Empirical rule states that exactly 68.27%, 95.45% and 99.73% of the values lie within one, two and three standard deviations of the mean, respectively for normal distribution.

A) answer is 68.27%

B) normal distribution is symmetric hencw percent of data within mean and 2 SD above the mean is $\frac{95.45}{2}=47.725$ %

C) The percentage of data towards the tail of 2 Standard deviation away from the mean is

$P(Z<-2)+P(Z>2)=1-0.9545=0.0455$

normal distribution is symmetric hence percentage of data that is more than 2 SD above the mean =$(\frac{0.0455}{2})100=2.28$%

12) $\mu =72,\sigma =8$

A) The percentage of these students scored more than 88 on the test is %

P(X >88) = P(Z > 98 - 12) = P(Z > 2) = 2.28 88 - 72

B) The percentage of these students scored less than 64 on the test is

$P(X<64)=P(Z<\frac{64-72}{8})=P(Z<-1)=\frac{1-0.6827}{2}=0.1587$ in percentage it is 15.87%

C)$P(56<X<72)=P(\frac{56-72}{8}<Z<\frac{72-72}{8})=P(-2<Z<0)$

$P(-2<Z<0)=P(Z<0)-P(Z<-2)=0.5-0.0228=0.4772$ in percentage 47.72%