Question

A random sample is selected from a population with mean μ = 100 and standard deviation σ = 10.

Determine the mean and standard deviation of the x sampling distribution for each of the following sample sizes. (Round the answers to three decimal places.)

(a) n = 8 μ = σ =

(b) n = 14 μ = σ =

(c) n = 34 μ = σ =

(d) n = 55 μ = σ =

(f) n = 110 μ = σ =

(e) n = 440 μ = σ =

A random sample is selected from a population with mean μ = 101 and standard deviation σ = 10. For which of the sample sizes would it be reasonable to think that the x sampling distribution is approximately normal in shape? (Select all that apply.)

n = 13

n = 16

n = 41

n = 60

n = 120

n = 480

An airplane with room for 100 passengers has a total baggage limit of 6000 lb. Suppose that the total weight of the baggage checked by an individual passenger is a random variable x with a mean value of 51 lb and a standard deviation of 23 lb. If 100 passengers will board a flight, what is the approximate probability that the total weight of their baggage will exceed the limit? (Hint: With n = 100, the total weight exceeds the limit when the average weight x exceeds 6000/100.) (Round your answer to four decimal places.)

A random sample is to be selected from a population that has a proportion of successes p = 0.69. Determine the mean and standard deviation of the sampling distribution of p̂ for each of the following sample sizes. (Round your standard deviations to four decimal places.)

(a)    n = 30

mean
standard deviation

(b)    n = 40

mean
standard deviation

(c)    n = 50

mean
standard deviation

(d)    n = 70

mean
standard deviation

(e)    n = 120

mean
standard deviation

(f)    n = 220

mean
standard deviation

An article reported that in a large study carried out in the state of New York, approximately 30% of the study subjects lived within 1 mile of a hazardous waste site. Let p denote the proportion of all New York residents who live within 1 mile of such a site, and suppose that p = 0.3.

(a) Would p̂ based on a random sample of only 10 residents have approximately a normal distribution? Explain why or why not.

Yes, because  np < 10 and n(1 − p) < 10.Yes, because  np > 10 and n(1 − p) > 10.     No, because np < 10.No, because np > 10.

(b) What are the mean value and standard deviation of p̂ based on a random sample of size 420? (Round your standard deviation to four decimal places.)

mean
standard deviation

(c) When n = 420, what is P(0.25 ≤ p̂ ≤ 0.35)?


(d) Is the probability calculated in Part (c) larger or smaller than would be the case if n = 520? Answer without actually calculating this probability.

The probability from Part (c) is smaller because as n increases, the standard deviation of p̂ decreases.The probability from Part (c) is larger because as n increases, the standard deviation of p̂ increases.     The probability from Part (c) is larger because as n increases, the standard deviation of p̂ decreases.The probability from Part (c) is smaller because as n increases, the standard deviation of p̂ increases.

Answer 1

Part a)

Part b)

Part c)

Part d)

Part e)

Part f)