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.
a) The expected value of the distribution of the sample mean (µx̄) can be determined using the formula:
µx̄ = µ
Since the average weight of students, faculty, and staff on campus is 150 lbs, the expected value of the sample mean weight is also 150 lbs.
b) The standard deviation of the sampling distribution of the sample mean (σx̄) can be calculated using the formula:
σx̄ = σ / (√n)
Where σ is the standard deviation of the population, and n is the sample size. In this case, the standard deviation of the population is given as 27 lbs, and the sample size is 16.
σx̄ = 27 / (√16)
σx̄ = 27 / 4
σx̄ = 6.75 lbs
So, the standard deviation of the sampling distribution of the sample mean weight is 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, we need to find the mean weight that corresponds to a total weight of 2500 lbs.
Total weight = Sample size * Sample mean
2500 = 16 * Sample mean
Sample mean = 2500 / 16
Sample mean ≈ 156.25 lbs
Any average weight for a sample of 16 people that is greater than 156.25 lbs will result in the total weight exceeding the weight limit of 2500 lbs.
d) To find the chance that a random sample of 16 people will exceed the weight limit, we need to calculate the probability of the sample mean weight being greater than the weight limit.
We can use the z-score formula to find the probability:
z = (X - µ) / (σ / √n)
Where X is the weight limit (2500 lbs), µ is the population mean (150 lbs), σ is the population standard deviation (27 lbs), and n is the sample size (16).
z = (2500 - 150) / (27 / √16)
z = 2350 / 6.75
Using a standard normal distribution table or a calculator, we can find the probability corresponding to this z-score. Let's assume it is 0.03 or 3%.
So, there is approximately a 3% chance that a random sample of 16 people will exceed the weight limit of 2500 lbs.
To answer these questions, we can use the concepts of sampling and the Central Limit Theorem.
a) The expected value of the distribution of the sample mean is equal to the population mean. Given that the average weight of students, faculty, and staff on campus is 150 lbs, the expected value of the sample mean is also 150 lbs.
b) The standard deviation of the sampling distribution of the sample mean (also known as the standard error) is determined by the population standard deviation and the sample size. Since the population standard deviation is given as 27 lbs and the sample size is 16, we can use the formula σ/√n, where σ is the population standard deviation and n is the sample size. Thus, the standard deviation of the sampling distribution is 27/√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, we need to calculate the maximum average weight possible. The maximum total weight can be found by multiplying the weight limit (2500 lbs) by the number of people (16). Therefore, the maximum total weight allowed is 2500 * 16 = 40,000 lbs. Divide this by the sample size (16) to find the maximum average weight: 40,000 / 16 = 2500 lbs. So, any average weight above 2500 lbs for a sample of 16 people will 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, we need to determine the probability of the sample mean being above the weight limit (2500 lbs). We can use the Central Limit Theorem, which states that the distribution of sample means will be normally distributed regardless of the shape of the population distribution, as long as the sample size is large enough. In this case, the sample size is 16, which is generally considered sufficient for the Central Limit Theorem to apply.
To find the probability, we need to convert the weight limit to a z-score and then find the corresponding probability using a standard normal distribution table or a statistical calculator. The z-score can be calculated using the formula: (sample mean - population mean) / (standard deviation / sqrt(sample size)). In this case, the population mean is 150 lbs, the standard deviation is 27 lbs, and the sample size is 16. Let's assume the sample mean is 160 lbs for the sake of example. The z-score would be: (160 - 150) / (27 / sqrt(16)) = 10 / (27 / 4) = 1.48.
Using a standard normal distribution table or a statistical calculator, you can find the probability associated with this z-score. For example, assuming a standard normal distribution table, you would look for the area to the right of 1.48 (since we are interested in the probability of exceeding the weight limit). The corresponding probability would be the remaining area, which you can convert to a percentage.
Alternatively, you can use a statistical calculator or software to find the probability directly.