Answer:
Approximately 16%
Explanation:
To solve this problem using the empirical rule, we need to first standardize the value of 170 pages using the mean and standard deviation provided:
z = (x - k) / o
where x is the value we want to find the probability for (170 pages), k is the mean (195 pages), and o is the standard deviation (25 pages).
So,
z = (170 - 195) / 25 = -1
Now, we can use the empirical rule, which states that for a normal distribution:
- About 68% of the data falls within 1 standard deviation of the mean
- About 95% of the data falls within 2 standard deviations of the mean
- About 99.7% of the data falls within 3 standard deviations of the mean
Since we know that the distribution is normal, and we want to find the probability that a randomly selected book has fewer than 170 pages (which is one standard deviation below the mean), we can use the empirical rule to estimate this probability as follows:
- From the empirical rule, we know that about 68% of the data falls within 1 standard deviation of the mean.
- Since the value of 170 pages is one standard deviation below the mean, we can estimate that the probability of randomly selecting a book with fewer than 170 pages is approximately 16% (which is half of the remaining 32% outside of one standard deviation below the mean).
Therefore, the probability that a randomly selected book has fewer than 170 pages is approximately 16%.