Question

A. Suppose x has a distribution with μ = 23 and σ = 15.

(a) If a random sample of size n = 39 is drawn, find μ_x, σ_x and P(23 ≤ x ≤ 25). (Round σ_x to two decimal places and the probability to four decimal places.)

μx =

σx =

P(23 ≤ x ≤ 25) =

(b) If a random sample of size n = 64 is drawn, find μ_x, σ_x and P(23 ≤ x ≤ 25). (Round σ_x to two decimal places and the probability to four decimal places.)

μx =

σx =

P(23 ≤ x ≤ 25) =

(c) Why should you expect the probability of part (b) to be higher than that of part (a)? (Hint: Consider the standard deviations in parts (a) and (b).)
The standard deviation of part (b) is  ---Select--- the same as larger than smaller than part (a) because of the  ---Select--- larger smaller same sample size. Therefore, the distribution about μ_x is  ---Select--- wider the same narrower

B. Suppose the heights of 18-year-old men are approximately normally distributed, with mean 67 inches and standard deviation 2 inches.

(a) What is the probability that an 18-year-old man selected at random is between 66 and 68 inches tall? (Round your answer to four decimal places.)


(b) If a random sample of twenty-nine 18-year-old men is selected, what is the probability that the mean height x is between 66 and 68 inches? (Round your answer to four decimal places.)


(c) Compare your answers to parts (a) and (b). Is the probability in part (b) much higher? Why would you expect this?

The probability in part (b) is much higher because the mean is larger for the x distribution.The probability in part (b) is much higher because the standard deviation is smaller for the x distribution.    The probability in part (b) is much lower because the standard deviation is smaller for the x distribution.The probability in part (b) is much higher because the mean is smaller for the x distribution.The probability in part (b) is much higher because the standard deviation is larger for the x distribution.

Answer 1

Solution: Given that μ = 23 and σ = 15 n = 39

(a) μx = μ = 23

σx = σ/sqrt(n) = 15/sqrt(39) = 2.40

P(23 ≤ x ≤ 25) = 0.2975

(b) for n = 64

μx = μ = 23

σx = σ/sqrt(n) = 15/sqrt(64) = 1.875 = 1.88 (rounded)

P(23 ≤ x ≤ 25) = 0.3569

(c) The standard deviation of part (b) is smaller than part (a) because of the larger sample size. Therefore, the distribution about μx is same.

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B. Given that mean = 67, sd = 2 inches

(a) P(66 < X < 68) = P((66-67)/2 < (X-mean)/sd < (68-67)/2)
= P(-0.5 < Z < 0.5)
= 0.3830

(b) forn n = 29
P(66 < X < 68) = P((66-67)/(2/sqrt(29)) < (X-mean)/(sd/sqrt(n)) < (68-67)/(2/sqrt(29))
= P(-2.6926 < Z < 2.6926)
= 0.9928

(c) option B. The probability in part (b) is much higher because the standard deviation is smaller for the x distribution.