Question

Consider an x distribution with standard deviation σ = 48.

(a) If specifications for a research project require the standard error of the corresponding x distribution to be 3, how large does the sample size need to be?
n =  

(b) If specifications for a research project require the standard error of the corresponding x distribution to be 1, how large does the sample size need to be?
n =

Suppose x has a distribution with μ = 27 and σ = 24.

(a) If a random sample of size n = 42 is drawn, find μ_x, σ_x and P(27 ≤ x ≤ 29). (Round σ_x to two decimal places and the probability to four decimal places.)

μx =

σx =

P(27 ≤ x ≤ 29) =

(b) If a random sample of size n = 62 is drawn, find μ_x, σ_x and P(27 ≤ x ≤ 29). (Round σ_x to two decimal places and the probability to four decimal places.)

μx =

σx =

P(27 ≤ x ≤ 29) =

(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 same smaller sample size. Therefore, the distribution about μ_x is  ---Select--- wider the same narrower .

Answer 1

Solution :

Given that standard deviation σ = 48

(a) standard error σ/sqrt(n) = 3

=> sqrt(n) = σ/3

=> n = (σ/3)^2

= (48/3)^2

= 256

=> sample size n = 256

(b) standard error σ/sqrt(n) = 1

=> sqrt(n) = σ/1

=> n = (σ/1)^2

= (48/1)^2

= 2304

======================================================

Solution :

Given that mean μ = 27 and standard deviation σ = 24

(a) when n = 42

=> μx = μ = 27

=> σx = σ/sqrt(n) = 24/sqrt(42) = 3.70

=> P(27 <= x <= 29) = P(26.5 < x < 29.5)

= P((26.5 - 27)/3.70 < (x - μx)/σx < (29.5 - 27)/3.70)

= P(-0.1351 < Z < 0.6757)

= 0.3074

(b) when n = 62

=> μx = μ = 27

=> σx = σ/sqrt(n) = 24/sqrt(62) = 3.05

=> P(27 <= x <= 29) = P(26.5 < x < 29.5)

= P((26.5 - 27)/3.05 < (x - μx)/σx < (29.5 - 27)/3.05)

= P(-0.1639 < Z < 0.8197)

= 0.3575

(c)

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