Question

Lisa has collected data to find that the number of pages per book on a book shelf has a normal distribution. What is the probability that a randomly selected book has fewer than 169 pages if the mean is 194 pages and the standard deviation is 25 pages? Use the empirical rule. Enter your answer as a percent rounded to two decimal places if necessary.

Answer 1

15.85%

Explanations:

According to the empirical rule for normal probability;

• 68% of given, data falls within 1σ of the mean

,

• 95%, of the given data falls within 2σ of the mean

,

• 99.7% of all data falls within 3σ of the mean

This rule shows an even distribution among the mean

From the data, we can say that 34% of the data is within μ±1σ

Given the following data

Mean = 194 pages

Standard deviation = 25

Hence 1 standard deviation of the mean will be 194 - 25 = 169 showing that 34% of data fall in that range

Since we need the probability that a randomly selected book has fewer than 169 pages, we use the normal distribution property which states:

50% of the data is left of the mean of which 34% fall within 1 standard deviation, hence the remaining would fall fewer than 169.

Pr(a randomly selected book has fewer than 169 pages) = 50% - 34% = 16%

To determine the final probability, we will subtract the half of 0.3(100-99.7) to have:

Pr(a randomly selected book has fewer than 169 pages) = 16 - 0.3/2

Pr(a randomly selected book has fewer than 169 pages) = 16 - 0.15

Pr(a randomly selected book has fewer than 169 pages) = 15.85%