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 165 pages if the mean is 192 pages and the standard deviation is 27 pages? Use the empirical rule. Enter your answer as a percent rounded to two decimal places if necessary.

Answer 1

Answer:

15.85%

Step-by-step explanation:

The empirical rule states that:

• 68% of all data falls within 1 standard deviation of the mean
• 95% of all data falls within 2 standard deviation of the mean
• 99.7% of all data falls within 3 standard deviation of the mean

When we say 1,2,3 standard deviation of the mean, that is, it is evenly distributed in both sides of the mean. For example:

68/2 = 34% of data falls within mean - 1*standard deviation, and

also, 34% of data falls within mean + 1*standard deviation

Now, we see that 192 - 27 = 165, so this is 1 standard deviation of the mean [mean - 1*standard deviation], so 34% of data fall in that range.

But we want, data that ARE FEWER than 165. So, we use property of normal distribution.

Half of data (50%) is left of mean, of which 34% fall within 1 standard deviation So the remaining would fall fewer than 165.

That is 50 - 34 = 16 - 0.15 = 15.85%

Note: since 99.7% fall within 3 standard deviation of mean, so 100 - 99.7 = 0.3 which is distributed in both sides, so 0.3/2=0.15 is subtracted