In the library on a university campus, there is a sign in the elevator that indicates a limit of 16 people. in addition there is a weight limit of 2500 lbs. assume that the average weight of students, faculty, and staff on campus is 150 lbs, that the standard deviation is 27 lbs and that the distribution of weights of individuals on campus is approsimately normal. if a random sample of 16 persons from the campus is to be taken.
a) what is the expected value of the distribution of the sample mean?
b)what si the standard deviation of the sampling distribution of the sample mean weight?
c)what average weights for a s ample of 16 people will result in the total weight exceeding the weight limit of 2500?
d)what is the chance that a random sample of 16 people will excedd the limit.
1.5
a) To find the expected value of the distribution of the sample mean, you will use the formula:
Expected Value = Population Mean
In this case, the population mean is 150 lbs. So the expected value of the distribution of the sample mean is also 150 lbs.
b) To find the standard deviation of the sampling distribution of the sample mean weight, you will use the formula:
Standard Deviation = Population Standard Deviation / sqrt(sample size)
In this case, the population standard deviation is 27 lbs, and the sample size is 16. So the standard deviation of the sampling distribution of the sample mean weight would be 27 / sqrt(16) = 6.75 lbs.
c) To find the average weights for a sample of 16 people that will result in the total weight exceeding the weight limit of 2500 lbs, you need to determine the maximum total weight allowed for a sample.
Maximum Total Weight = Weight Limit of Elevator * Sample Size
Maximum Total Weight = 2500 lbs * 16 = 40000 lbs
Now, let's assume the average weight of the sample is X. The total weight of the sample would then be X * 16.
So, the equation would be:
X * 16 > 40000
X > 40000 / 16
X > 2500 lbs
Therefore, any average weight for a sample of 16 people that is greater than 2500 lbs would result in the total weight exceeding the weight limit.
d) To calculate the chance that a random sample of 16 people will exceed the weight limit, you will need to use the normal distribution.
First, calculate the z-score for the weight limit:
z = (Weight Limit - Population Mean) / (Population Standard Deviation / sqrt(sample size))
z = (2500 - 150) / (27 / sqrt(16))
z = 2350 / 6.75
z ≈ 348.15
Since this z-score is extremely large, the probability of having a random sample mean exceeding the weight limit of 2500 lbs is extremely close to 1 or 100%.