Finding the probability of something from the population

Question

Based on a health care researcher of a hospital, the height of female individuals aged 25-35 is normally distributed with a mean of 161 cm and a standard deviation of 9 cm. Please:

1. Determine with appropriate calculations if you agree or disagree with the statement that “Less than 5% of the population items lies beyond 2 standard deviations from the population mean.”
2. Calculate the probability that an individual in the population (X) is more than 5 standard deviations bigger than the population mean.
3. Calculate the amount for the top and bottom 19th percentage of the population
4. Calculate the probability that an individual in the population (X) is more than 5 standard deviations bigger than the population mean. Explain if it is possible to find such an individual in the population in reference to the scenario.

Answer 1

Per the empirical rule 95% of the data would be within 2 standard deviations of the mean, so %5 of the data would be beyond two standard deviations from the mean.

The probability that a value would be 5 standard deviations beyond the mean would be P(z > 5) is basically 0.

Bottom 19 percentile. z-sore for 0.19 is -0.878.

use formula z = (x - mean)/SD, where x is height mean is 161 and SD is 9 and z is -0.878. Solve for x and that is the 19th percentile.

For top 19th percentile (or 81st percentile), you will repeat the above calculation, except z in this case willbe 0.878 (as the normal distribution is symmetric).

for individual beyond 5 SD, z = 5

5 = (x -161)/6

x = 191