Question

What is the probability that in a new sample of 400 telephone numbers, at least 150 individuals will cooperate and respond to the questions? In other words, what isthe probability that the sample proportion will be at least 150/400 = .375?

Calculate the probability to 4 decimals.

Answer 1

Part 1

The probability that the sample proportion will be at least 0.375 is 0.8461.

Part 2.a

The standard error of the mean is 0.46.

Part 2.b

The margin of error, with 95% confidence is 0.89.

Part 3

The sample size is 199.

Answer 2

1.
p0 = 0.4
x = 150 , n = 400 , p1=x/n = 0.375
.
standard error, SE = sqrt(p0(1-p0)/n)
SE = sqrt(0.4(1-0.4) / 400) = 0.0245
.
test statistic, z = (p1-p0) / SE
z = (0.375-0.4) / 0.0245 = -1.02

probability that the sample proportion will be at least 150/400 = .375
= P(Z>-1.02)
= P(Z<1.02)
= 0.8461
____________________________________________________________
2.
sample size,n = 77
sample mean,x = 14
standard deviation,sigma = 4

a.
standard error of the mean = sigma/sqrt(n)
= 4/sqrt(77)
= 0.46 (answer may be 0.45)

b.
alpha,a = 1-0.95 = 0.05
Za/2 = Z0.025 = 1.96

margin of error = Za/2 * (sigma/sqrt(n))
= 1.96 * (4/sqrt(77))
= 0.89 (answer may be 0.90)
____________________________________________________________-

3.
error,E = (101.5-98)/2 = 1.75
standard deviation,sigma = 15
alpha,a = 1-0.90 = 0.10

Za/2 = Z0.05 = 1.645
sample size,n >= [Za/2 *(sigma/E)]^2
n >= [1.645 * (15/1.75)] ^2
n >= 198.81
n = 199