Question

Which of the following statements is not a property of the normal probability distribution?

A. The normal distribution is symmetric
B. Exactly two-thirds of all possible observed values of the random variable x are within plus or minus one standard deviations of the population mean
C. The mean, median and mode are equal
D. The area under the normal curve to the right of the mean is equal to the area under the normal curve to the left of the mean
E. All of the above answers are properties of the normal distribution

Answer 1

The correct option is: \(\mathrm{B}\)

Exactly two-thirds of all possible observed values of the random variable \(x\) are within plus or minus one standard deviation of the population mean.

Explanation:

The two-thirds \(\left(\frac{2}{3}=0.6667 \Rightarrow 66.67 \%\right)\) of all possible observed values of the random variable \(x\) in not within plus or minus one standard deviation of the population mean.

From the Empirical rule:

Hence, the one standard deviation of the population mean is \(68 \%\).

Answer 2

: E. All of the above answers are properties of the normal distribution

Answer 3

SOLUTION :

Option B. (ANSWER).

Explanation :

Area under normal curve between -1 and + 1 SD = 0.6826

And 2/3 rd = 0.6667 . 

Hence,  within +1 and - 1 SD, it is not exactly 2/3rd but slightly more than that.