Question

A certain data distribution has a mean of 18 and a standard deviation of 3

Wha, is the value would have a Z-score of -3.2?

Assuming that the distribution is normal, use the Empirical Rule to determine what proportion of this distribution would be found to be between 15 and 24

Assuming that the distribution is normal, use the Empirical Rule to determine what proportion of his distribution would be found to be between 12 and 15

Now imagine that this distribution is NOT guaranteed to be normally distributed. What would be the minimum proportion of this distribution that might be found between 13.5 and 22.5?

Answer 1

Given that, mean = 18 and standard deviation = 3

A) The value for z-score of -3.2 is,

x = (Z * sd) + mean = (-3.2 * 3) + 18 = -9.6 + 18 = 8.4

Required value is 8.4

According to e rule,

i) Approximately 68% of the data fall within 1 standard deviations of the mean.

ii) Approximately 95% of the data fall within 2 standard deviations of the mean.

iii) Approximately 99.7% of the data fall within 3 standard deviations of the mean.

B) mean - sd = 18 - 3 = 15 and mean + 2 * sd = 18 + (2 * 3) = 24

Therefore, approximately, 81.5% of this distribution would be found to be between 15 and 24.

C) mean - 2 * sd = 18 - (2 * 3) = 12 and mean - sd = 18 - 3 = 15

Therefore, approximately, 13.5% of this distribution would be found to be between 13 and 15.

D) if given distribution is NOT guaranteed to be normally distributed, then we should used Chebyshev's theorem to calculate the proportion between two values.

mean - 1.5 * sd = 18 - (1.5 * 3) = 13.5 and

mean + 1.5 * sd = 18 + (1.5 * 3) = 22.5

[1 - (1/(1.5)²) ] * 100 = 0.5556 * 100 = 55.56%

Therefore, at least 55.56% proportion of this distribution that might be found between 13.5 and 22.5