A survey indicates that 42% of consumers from a certain region are willing to sp
ID: 3126692 • Letter: A
Question
A survey indicates that 42% of consumers from a certain region are willing to spend extra for products and services from socially responsible companies. The researchers define these consumers as socially conscious consumers. According to one of the researchers, marketers need to know who these consumers are if they want to maximize the social and business return of their cause-marketing efforts. Suppose a sample of 75 consumers from the region is selected. If possible, use the normal distribution to approximate the sampling distribution of the proportion. Complete parts (a) through (F) below. Explain step by step please.
A. What is the probability that in the sample fewer than 42% are willing to spend extra for products and services from socially responsible companies?
B. What is the probability that in the sample between 35% and 49% are willing to spend extra for products and services from socially responsible companies?
C. What is the probability that in the sample more than 35% are willing to spend extra for products and services from socially responsible companies?
D. If a sample of 300 is taken, what is the probability that in the sample fewer than 42% are willing to spend extra for products and services from socially responsible companies?
e. If a sample of 300 is taken, what is the probability that in the sample between 35% and 49% are willing to spend extra for products and services from socially responsible companies?
f. If a sample of 300 is taken, what is the probability that in the sample more than 35% are willing to spend extra for products and services from socially responsible companies?
Explanation / Answer
A)
We first get the z score for the critical value. As z = (x - u) / s, then as
x = critical value = 0.42
u = mean = p = 0.42
s = standard deviation = sqrt(p(1-p)/n) = 0.056991227
Thus,
z = (x - u) / s = 0
Thus, using a table/technology, the left tailed area of this is
P(z < 0 ) = 0.5 [ANSWER]
****************
b)
Here,
n = 75
p = 0.42
We first get the z score for the two values. As z = (x - u) / s, then as
x1 = lower bound = 0.35
x2 = upper bound = 0.49
u = mean = p = 0.42
s = standard deviation = sqrt(p(1-p)/n) = 0.056991227
Thus, the two z scores are
z1 = lower z score = (x1 - u)/s = -1.228259211
z2 = upper z score = (x2 - u) / s = 1.228259211
Using table/technology, the left tailed areas between these z scores is
P(z < z1) = 0.109674839
P(z < z2) = 0.890325161
Thus, the area between them, by subtracting these areas, is
P(z1 < z < z2) = 0.780650322 [ANSWER]
*******************
c)
We first get the z score for the critical value. As z = (x - u) / s, then as
x = critical value = 0.35
u = mean = p = 0.42
s = standard deviation = sqrt(p(1-p)/n) = 0.056991227
Thus,
z = (x - u) / s = -1.228259211
Thus, using a table/technology, the right tailed area of this is
P(z > -1.228259211 ) = 0.890325161 [ANSWER]
*******************
d)
We first get the z score for the critical value. As z = (x - u) / s, then as
x = critical value = 0.42
u = mean = p = 0.42
s = standard deviation = sqrt(p(1-p)/n) = 0.028495614
Thus,
z = (x - u) / s = 0
Thus, using a table/technology, the left tailed area of this is
P(z < 0 ) = 0.5 [ANSWER]
********************
e)
Here,
n = 300
p = 0.42
We first get the z score for the two values. As z = (x - u) / s, then as
x1 = lower bound = 0.35
x2 = upper bound = 0.49
u = mean = p = 0.42
s = standard deviation = sqrt(p(1-p)/n) = 0.028495614
Thus, the two z scores are
z1 = lower z score = (x1 - u)/s = -2.456518422
z2 = upper z score = (x2 - u) / s = 2.456518422
Using table/technology, the left tailed areas between these z scores is
P(z < z1) = 0.00701453
P(z < z2) = 0.99298547
Thus, the area between them, by subtracting these areas, is
P(z1 < z < z2) = 0.985970939 [ANSWER]
*******************
f)
We first get the z score for the critical value. As z = (x - u) / s, then as
x = critical value = 0.35
u = mean = p = 0.42
s = standard deviation = sqrt(p(1-p)/n) = 0.028495614
Thus,
z = (x - u) / s = -2.456518422
Thus, using a table/technology, the right tailed area of this is
P(z > -2.456518422 ) = 0.99298547 [ANSWER]
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