Question

For bone density scores that are normally distributed with a mean of 0 and a standard deviation of​ 1, find the percentage of scores that are

a. significantly high​ (or at least 2 standard deviations above the​ mean).

b. significantly low​ (or at least 2 standard deviations below the​ mean).

c. not significant​ (or less than 2 standard deviations away from the​ mean).

Answer 1

Concepts and reason

The normal distribution is a continuous probability distribution that is symmetrical on both sides of the mean and any particular normal distribution is completely specified by the numbers mean and standard deviation.

Fundamentals

The formula of standard normal score is,

Here, is the population mean and is the population standard deviation of the normal distribution.

The MS Excel function of to calculate the normal probability value is,$(=NORMSDIST(Z)) $

a)

The available information is shown below,

Let denote the bone density score.

Calculate the percentage that bone density score is significantly high or at least 2 standard deviation above the mean.

b)

Calculate the percentage that bone density score is significantly low or at least 2 standard deviation below the mean.

c)

Calculate the percentage that bone density score is not significant or at less than 2 standard deviation away the mean.

Ans: Part a

The percentage that bone density score is significantly high or at least 2 standard deviation above the mean is 2.28%

Part b

The percentage that bone density score is significantly low or at least 2 standard deviation below the mean is 2.28%

Part c

The percentage that bone density score is not significant or at less than 2 standard deviation away the mean is 95.44%

Answer 2

(a)

Significantly high = P(z > 2) = 0.0228 = 2.28%

(b)

Significantly low = P(z < -2) = 0.0228 = 2.28%

(c)

Not significant = 100 - 2 * 2.28 = 95.44%