Questions: 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 (μ) is 189 pages and the standard deviation (σ) is 20 pages? Use the empirical rule. Enter your answer as a percent rounded to two decimal places if necessary. Provide your answer below: %

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 (μ) is 189 pages and the standard deviation (σ) is 20 pages? Use the empirical rule. Enter your answer as a percent rounded to two decimal places if necessary.

Provide your answer below: %

Transcript text: 3/25 QUESTION 4 - 1 POINT 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 $(\mu)$ is 189 pages and the standard deviation $(\sigma)$ is 20 pages? Use the empirical rule. Enter your answer as a percent rounded to two decimal places if necessary. Provide your answer below: $\square$ $\%$

Solution

Solution Steps

Step 1: Calculate the Z-score

To find the probability of a value being less than 169, we first calculate the Z-score using the formula: \[Z = \frac{X - \mu}{\sigma}\] Substituting the given values, we get: \[Z = \frac{169 - 189}{20} = -1\]

Step 2: Estimate the probability using the empirical rule

Using the empirical rule, since |Z| < 2, approximately 95% of the data lies within 2 standard deviations of the mean. Therefore, the probability that a randomly selected item has a value less than 169 is estimated to be: \[47.5\%\]