Question

It is known that the height (X) of females in a certain country is normally distributed
with mean μ = 1600 millimeters (mms) and standard deviation σ = 55 mms.

Use the normal distribution to estimate the 88th percentile of this population,
i.e. find the cutpoint " k " so that percent population at most " k " mms tall is 88 percent.

Answer to one decimal place.

Answer 1

Answer 2

To find the 88th percentile of the population height distribution, we need to determine the height value "k" below which 88% of the population falls. In other words, we are looking for the height value that separates the lower 88% of the population from the upper 12%.

The z-score formula for a normal distribution is:

z = (X - μ) / σ

where: X = the height value we want to find μ = mean height of the population (1600 mm) σ = standard deviation of the population (55 mm)

Since we are looking for the 88th percentile, we need to find the z-score corresponding to the cumulative probability of 0.88. We can use a standard normal distribution table or a calculator to find this z-score.

Using a standard normal distribution table or calculator, the z-score for a cumulative probability of 0.88 is approximately 1.175.

Now, we can rearrange the z-score formula to solve for the height value "X":

X = μ + (z * σ) X = 1600 + (1.175 * 55) X ≈ 1600 + 64.63 X ≈ 1664.6 mm

So, the 88th percentile of the population height distribution is approximately 1664.6 millimeters (mm).