Question

You may need to use the appropriate appendix table or technology to answer this question. Suppose we are interested in the Annual Salary of Electronics Associated, Inc. (EAI) managers and we find that for a sample of n = 30 managers, there was 0.5034 probability of obtaining a sample mean within ± $500 of the population mean (see the figure below). For parts (a) and (b), let the sample mean be 51,800 and σ = 4,000. (Round your answers to four decimal places.) (a) What is the probability that x is within ±$500 of the population mean if a sample of size 40 is used? (b) Answer part (a) for a sample of size 80. The title of the diagram is "Sampling Distribution of x". A bell-shaped curve divided into three areas is above a horizontal axis labeled x. The text sigma sub x bar = 730.30 is on the figure. The horizontal axis has three tick marks. In the order they appear, from the left side of the figure to the right, they are: 51,300, 51,800, and 52,300. The label 51,800 is below the maximum point on the curve and in the center of the horizontal axis. The first area under the curve is to the left of 51,300, is shaded, and is labeled P(x < 51,300). The second area under the curve is between 51,300 and 52,300, is shaded, and is labeled P(51,300 ≤ x ≤ 52,300). The third area under the curve is to the right of 52,300 and is shaded. There is no label.

Answer 1

a)

for normal distribution z score =(X-μ)/σx
here mean=       μ=	51800
std deviation   =σ=	4000.000
sample size       =n=	40
std error=σx̅=σ/√n=	632.4555

probability =

P(51300<X<52300)

=

P(-0.79<Z<0.79)=

0.7852-0.2148=

0.5704

b)

sample size       =n=	80
std error=σx̅=σ/√n=	447.2136

probability =

P(51300<X<52300)

=

P(-1.12<Z<1.12)=

0.8686-0.1314=

0.7372