Question

When answering the following questions, use appropriate notation and units when necessary. Round all

proportions / probabilities to 4 decimals, and x-values to the nearest whole number, and all other figures to
the accuracy specified. For Normal Distribution problems b) – f): draw a picture, write down calculator
function with inputs, and answer with correct rounding.
1) The number of times a person in the United States moves homes in his or her lifetime is normally
distributed with a mean of 12 and a standard deviation of 4.
a) [4 pts] What would the sampling distribution for the sample mean be for samples of 9 people from the
United States? In other words, what is the mean ( ), standard deviation (), and shape of the
distribution. Round to one decimal when needed.
µx σ x b) [4 pts] If a person from the United Sates is chosen at random, what is the probability he/she will have
moved at most 5 times?
c) [4 pts] 85% of people from the United Sates will have moved at least ____ times in their lifetime.

d) [4 pts] If 9 people from the United Sates are chosen at random, what is the probability their mean
number of moves per lifetime is greater than 17?
e) [4 pts] If 9 people from the United Sates are chosen at random, what is the probability their mean
number of moves per lifetime is less than 9?
f) [4 pts] For a random sample of 9 people from the United Sates, the middle 50% of sample means is
between _____ and _____ number of moves per lifetime. Note these values would have to be Q1 and Q3.

Answer 1

a) The sampling distribution for the sample mean would be a normal distribution with a mean equal to the population mean (12) and a standard deviation equal to the population standard deviation divided by the square root of the sample size (4 / sqrt(9) = 2).

So, the mean of the sampling distribution would be 12, the standard deviation would be 2, and the shape would be normal.

b) We can use a standard normal table or a normal calculator function to find the probability of a person having moved at most 5 times. We need to first standardize the random variable by subtracting the population mean (12) and dividing by the population standard deviation (4):

z = (5 - 12) / 4 = -2.5

Using a standard normal table or a normal calculator function, we find that the probability of a person having moved at most 5 times is 0.0062.

c) To find the value such that 85% of the people will have moved at least that many times, we need to find the 85th percentile of the normal distribution. We can use a normal calculator function to find the inverse standard normal value for 0.85. Let x be the number of moves.

z = (x - 12) / 4

x = 12 + 4 * invNorm(0.85)

Using a normal calculator function, we find that invNorm(0.85) = 0.841. So,

x = 12 + 4 * 0.841 = 16.164

Rounding to the nearest whole number, we find that 85% of people from the United States will have moved at least 16 times in their lifetime.

d) To find the probability that a random sample of 9 people will have a mean number of moves greater than 17, we need to standardize the sample mean:

z = (17 - 12) / (2) = 2.5

Using a standard normal table or a normal calculator function, we find that the probability of a standard normal random variable being greater than 2.5 is 0.0062.

So, the probability that a random sample of 9 people will have a mean number of moves greater than 17 is 0.0062.

e) To find the probability that a random sample of 9 people will have a mean number of moves less than 9, we need to standardize the sample mean:

z = (9 - 12) / (2) = -1.5

Using a standard normal table or a normal calculator function, we find that the probability of a standard normal random variable being less than -1.5 is 0.0668.

So, the probability that a random sample of 9 people will have a mean number of moves less than 9 is 0.0668.

f) To find the middle 50% of sample means, we need to find the 25th and 75th percentiles of the normal distribution of sample means. We can use a normal calculator function to find the inverse standard normal values for 0.25 and 0.75. Let x be the number of moves.

z = (x - 12) / (2)

x = 12 + 2 * invNorm(0.25) or x = 12 + 2 * invNorm(0.75)

Using a normal calculator function, we find that invNorm(0.25) = -0.675 and invNorm(0.75) = 0.675. So,

x = 12 + 2 * -0.675 = 11.35 or x = 12 + 2 * 0.675 = 13.