Times for an ambulance to respond to a medical emergency in a certain town are normally distributed with a mean of 450 seconds and a standard deviation of 50 seconds.

Suppose there are 160 emergencies in that town.

In about how many emergencies are the response times expected between 400 seconds and 500 seconds?



51

55 <my choice

80

109

use first choice, area from values

mean 450
sd 50

between 400 and 500

output is .6827 or 68% are between
well that is a little less thsan .7*160 = 112
so
109

http://davidmlane.com/hyperstat/z_table.html

how do I use that ?

To find the number of emergencies where the response times are expected to be between 400 seconds and 500 seconds, we need to determine the probability of an emergency falling within this range. Since we know the response times follow a normal distribution with a mean of 450 seconds and a standard deviation of 50 seconds, we can use the cumulative distribution function (CDF) of the normal distribution to find this probability.

The CDF gives us the probability of a random variable (in this case, response time) being less than or equal to a certain value. We can calculate this using the z-score formula:

z = (x - μ) / σ

Where:
z is the z-score
x is the value we want to calculate the probability for (500 seconds in this case)
μ is the mean (450 seconds)
σ is the standard deviation (50 seconds)

Calculating the z-score for x = 500 seconds:

z = (500 - 450) / 50
z = 50 / 50
z = 1

Next, we can use a standard normal distribution table (also known as a Z-table) or a calculator to find the probability corresponding to this z-score. The probability corresponds to the area under the curve between the mean and the given value.

Looking up the z-score of 1 in the Z-table, we find that the probability is approximately 0.8413.

Now we need to calculate the probability for x = 400 seconds. Using the same formula:

z = (400 - 450) / 50
z = -50 / 50
z = -1

Looking up the z-score of -1 in the Z-table, we find that the probability is approximately 0.1587.

To find the probability of an emergency falling within the range of 400 seconds and 500 seconds, we subtract the probability for x = 400 seconds from the probability for x = 500 seconds:

0.8413 - 0.1587 = 0.6826

So, there is a 0.6826 probability that an emergency's response time will fall between 400 seconds and 500 seconds. To find the expected number of emergencies within this range, we multiply this probability by the total number of emergencies:

0.6826 * 160 = 109

Therefore, the expected number of emergencies with response times between 400 seconds and 500 seconds is 109.