Statistics question, check description

Question

A sample of 300 adult women finds that their heights are approximately Normally distributed with an average height of 65 inches and a standard deviation of 2.5 inches.

1) Shuri is 69.5 inches tall. What is her standard score for height?

2) How many women in the sample would you expect to be less than 62 inches tall?

3) How tall or short would a woman have to be to be considered an outlier? Show your work.

Answer 1

1) To find a standardized score (z-score), take the individual core minus the mean, then divide by the standard deviation: z = (score - mean)/ SD

Shuri's standardized score would be: z = (69.5 - 65)/2.5 = 1.8

2) First, find the z-score for 62 in: z= (62 - 65) / 2.5 = -1.2

Next, look up -1.2 in a z-table. The corresponding number in the table is .1151.

Multiple this number by 100 to get a percentage: .1151 * 100 = 11.51%

Thus 11.51% of people will be less than 62 inches.

3) Outliers are defined differently dependent on who you ask, often values that are 3 standard deviations (or more) above or below the mean are considered an outlier.

Use the same formula for finding a z-score: z= (score - mean)/ SD

For the taller height use 3:

3= (x - 65)/2.5

Do a little algebra and you get: 3(2.5) + 65 = x thus x = 72.5 or taller

For the shorter height, use -3:

-3(2.5) + 65 = x thus x = 57.5 or shorter

If you need to use a different value for an outlier, then use that number in place of +/- 3.