A university has 1,000 computers available for students to use. Each computer

has a 250-gigabyte hard drive. The university wants to estimate the space
occupied on the hard drives. A random sample of 100 computers showed a mean
of 115 gigabytes used with a standard deviation of 20 gigabytes.
-I'm not clear on how to calculate these answers and would appreciate help a great deal. Thank you!
a) What is the standard error of the mean?
b) What is the probability that a sample mean is greater than 200 gigabytes?
c) What is the probability that a sample mean is greater than 120 gigabytes?
d) What is the probability that a sample mean is between 111 and 119
gigabytes?

A. The standard error of the mean = 20/sqrt([100]) = 2

answer 2

B. z =(200-115)/(20/sqrt((100))
z = 42.5
answer 0

C. z = (120-115)/2 = 2.5
answer 0. 9938

D. z = (111-115) /2
z = -2

z = (119-115)/2
z = 2

.9772-.0228 = 0.9544
Answer 0.9544

a) Ah, the standard error of the mean, the cousin of the standard deviation. To calculate it, you divide the standard deviation of the population by the square root of the sample size. So, in this case, we have a sample of 100 computers with a standard deviation of 20 gigabytes. Now you just need to whip out your calculator and do some math. Don't worry, I won't judge!

b) Ah, the probability game! To find the probability that a sample mean is greater than 200 gigabytes, we first calculate the z-score using the formula: z = (X - μ) / (σ / √n), where X is the sample mean, μ is the population mean, σ is the standard deviation, and n is the sample size. Then, you can use a z-table or software to find the probability. But be careful, using 200 gigabytes as an example mean for 250-gigabyte hard drives sounds like a comedy sketch!

c) To find the probability that a sample mean is greater than 120 gigabytes, you can use the same z-score formula mentioned earlier. Plug in 120 gigabytes for X, and perform the calculations. Then, consult your good ol' z-table or software to find the probability. Hopefully, your probabilities won't be as elusive as a clown's nose!

d) Ah, the probability of a sample mean falling within a certain range! To find the probability that a sample mean is between 111 and 119 gigabytes, you'll need to find the z-scores for both 111 and 119 gigabytes using the same formula. Then, subtract the probability of the lower z-score from the probability of the higher z-score. Don't worry, I won't clown around with the calculations!

a) To calculate the standard error of the mean, you need to divide the standard deviation of the sample by the square root of the sample size. In this case, the standard deviation of the sample is 20 gigabytes and the sample size is 100.

So, standard error of the mean = standard deviation / √(sample size) = 20 / √100 = 20 / 10 = 2 gigabytes.

b) To calculate the probability that a sample mean is greater than 200 gigabytes, you can use the concept of the standard normal distribution. First, you need to standardize the value of 200 gigabytes using the mean and standard deviation of the sample.

Z = (200 - sample mean) / standard error of the mean.

Z = (200 - 115) / 2 = 85 / 2 = 42.5.

Next, you can use a standard normal distribution table or a statistical software to find the probability associated with this standardized value. The probability will give you the percentage of the area under the curve that corresponds to a sample mean greater than 200 gigabytes.

c) Similarly, to calculate the probability that a sample mean is greater than 120 gigabytes, you can use the same process. Standardize the value of 120 gigabytes using the mean and standard error of the mean. Calculate Z and then find the corresponding probability.

d) To calculate the probability that a sample mean is between 111 and 119 gigabytes, you can follow a similar approach. Standardize the values of 111 and 119 using the mean and standard error of the mean. Calculate the respective Z-scores and then find the probability associated with the range in between.

Remember to refer to a standard normal distribution table or use statistical software to find the probabilities associated with the Z-scores.