A fitness company surveyed the weights of its 90 costumers. Assume that the weights are normally distrubted with a mean of 162 lbs and a standard deviation of 3.5 lbs. If you select one of its costumers randomly, what is the probability that the costumer's weight is

a) more than 162 lbs?
b) more than 155 lbs?
c) less than 172.5 lbs?
d) between 158.5 lbs and 169 lbs?

a) 50%

b) 97.7%
c) 99.9%
d) 81.9%

These answers were obtained using the normal disribution calculation tool at
http://psych.colorado.edu/~mcclella/java/normal/accurateNormal.html

I did not make use of the fact that the mean and standard deviation were obtained from 90 samples. I assumed 162 was the true mean and 3.5 was the true std deviation.

To find the probability for each case, we need to use the standard normal distribution, also known as the Z-distribution. We can convert the raw scores (weights in this case) to Z-scores by subtracting the mean and dividing by the standard deviation. The Z-scores will then be used to find the probabilities.

Let's calculate the probabilities for each case:

a) To find the probability that the customer's weight is more than 162 lbs, we need to find the area under the curve to the right of the Z-score for 162 lbs.

First, calculate the Z-score:
Z = (162 - mean) / standard deviation
Z = (162 - 162) / 3.5
Z = 0

Now, find the probability using the Z-table or calculator:
P(Z > 0) = 0.5 - 0.5 = 0.5

So the probability that the customer's weight is more than 162 lbs is 0.5.

b) To find the probability that the customer's weight is more than 155 lbs, follow the same process:
Z = (155 - mean) / standard deviation
Z = (155 - 162) / 3.5
Z = -2

P(Z > -2) can be found using the Z-table or calculator:
P(Z > -2) = 0.9772

So the probability that the customer's weight is more than 155 lbs is 0.9772.

c) To find the probability that the customer's weight is less than 172.5 lbs, follow the same process:
Z = (172.5 - mean) / standard deviation
Z = (172.5 - 162) / 3.5
Z = 3

P(Z < 3) can be found using the Z-table or calculator:
P(Z < 3) = 0.9987

So the probability that the customer's weight is less than 172.5 lbs is 0.9987.

d) To find the probability that the customer's weight is between 158.5 lbs and 169 lbs, we need to find the area under the curve between the Z-scores for 158.5 lbs and 169 lbs.

First, calculate the Z-scores:
Z1 = (158.5 - mean) / standard deviation
Z1 = (158.5 - 162) / 3.5
Z1 = -1
Z2 = (169 - mean) / standard deviation
Z2 = (169 - 162) / 3.5
Z2 = 2

Now, find the probability:
P(-1 < Z < 2) can be found using the Z-table or calculator:
P(-1 < Z < 2) = P(Z < 2) - P(Z < -1) = 0.9772 - 0.1587 = 0.8185

So the probability that the customer's weight is between 158.5 lbs and 169 lbs is 0.8185.