normal distribution

Question

Suppose that the duration of a particular type of criminal trial is known to be normally distributed with a mean of 25 days and a standard deviation of 6 days.

72% of all of these types of trials are completed within how many days? (Round your answer to two decimal places.)

Answer 1

To find the number of days within which 72% of the trials are completed, we need to determine the corresponding z-score and then convert it back to the actual number of days using the mean and standard deviation.

First, we find the z-score corresponding to the 72nd percentile (since the normal distribution is symmetrical):

Percentile = 0.72

Area under the curve to the left of the z-score = Percentile = 0.72

Using a standard normal distribution table or a calculator, we can find the z-score that corresponds to an area of 0.72. The closest z-score is approximately 0.675.

Now, we can convert the z-score back to the actual number of days:

z = (x - mean) / standard deviation

0.675 = (x - 25) / 6

0.675 * 6 = x - 25

4.05 + 25 = x

x ≈ 29.05

Rounded to two decimal places, approximately 72% of all these types of trials are completed within 29.05 days. Therefore, we can round this to 29.05 days or simply 29 days.

Answer 2

18.52 to 31.48 days is the range of days for the criminal trials to end 72% of the time

25 days mean with sd = 6 days

z score = (25-x)/6 = z, find z corresponding to .72

.72 is slightly more than .68 = 68%. Using the empirical rule 68% of the data occurs within 1 standard deviation from the mean 72% = .72 is slightly more than 1 standard deviation.

using 68%, 25-x = 6

x = 25-6= 19

72% is slightly more than 68% so expect the lower bound to be slightly less than 19, such as 18.52

find that using z tables

similarly the upper bound is (25+(25-18.52)) = 50-18.52 = 31.48

72% fall between 18.52 and 31.48

but if you want a point estimate, one number of days, it's the mean = 25 days

which is the average of the endpoints (18.52+31.48)/2 = 50/2 = 25 days

but the question is slightly ambiguous and may be asking for how many days are 72% of the cases completed, meaning when are 28% not completed, with the 28% on the right tail of the normal distribution, not split into two tails. Then find the z score corresponding to z = (x-25)/6 with an area under the normal curve = 72.-.5 = .22

.22 x 2 = .44 corresponding to x = 28.50

(28.498-25)/6 = .583= z

68% corresponds to 1 sd

28% on the right tail corresponds to z= .583

28% are not completed within 28 1/2 days

72% are completed within 28.50 days