Question

Times for an ambulance to respond to a medical emergency in a certain town are normally distributed with a mean of 400 seconds and a standard deviation of 50 seconds.

Suppose there are 144 emergencies in that town.

In about how many emergencies are the response times expected between 350 seconds and 450 seconds?
49
51
96
98

Answer 1

98 emergencies

Explanation:

First, we analyze the data given, if the mean is 400 seconds, and the standard deviation is 50 seconds.

So, the interval given of 350-450 corresponds to -1 and 1 standard deviations on a normal distribution, which means that: Z = ±1

interval = 400±50 = 400 ± 1 standard deviation

We look at a Z probability distribution table for these values of Z and have:

Z(-1)= 0.1587 and Z (1)= 0.8413

Then we take these values and subtract them, from largest to smallest, in that way, we find the probability inside that range (±1 standard deviation),

0.8413-0.1587 = 0.6826

Finally, multiply this probability with the number of emergencies (144)

0.6829*144= 98.2944 which approximates to 98 emergencies inside the 350 to 450 seconds

Answer 2

Explanation:

In the normal distribution curve, the mean is in the middle and each line to the left and to the right of that mean represent 1- and 1+ the standard deviation. If our mean is 400, then 400 + 50 = 450; 450 + 50 = 500; 500 + 50 = 550. Going from the mean to the left, we subtract the standard deviation and 400 - 50 = 350; 350 - 50 = 300; 300 - 50 = 250. We are interested in the range that falls between 350 and 450 as a percentage. That range represents the two middle sections, each containing 34% of the data. So the total percentage of response times is 68%. We are looking then for 68% of the 144 emergency response times in town. .68(144) = 97.92 or 98 emergencies that have response times of between 350 and 450 seconds.