1) According to a survey in a country, 37% of adults do not have any credit card
ID: 3125787 • Letter: 1
Question
1) According to a survey in a country, 37% of adults do not have any credit cards. Suppose a simple random sample of 400 adults is obtained.
A) Describe the sampling distribution of p^, the sample proportion of adults who do not have a credit card. Choose the phrase that best describes the shape of the sampling distribution of p^ below:
a) Approximately normal because n <= 0.05N and np(1-p) < 10.
b) Not normal because n <= 0.05 and np(1-p) >= 10.
c) Not normal because n <= 0.05 and np(1-p) < 10.
d) Approximately normal because n <= 0.05N and np(1-p) >= 10.
B) Determine the mean of the sampling distribution of p^ (round to two decimal places as needed).
C) Determine the standard deviation of the sampling distribution of p^ (round to three decimal places as needed).
D) In the random sample of 400 adults, what is the probability that less than 34% have no credit cards? (round to four decimal places as needed).
E) Would it be unusual if a random sample of 400 adults results in 164 or more having no credit cards?
a) The result is not unusual because the probability that p^ is greater than or equal to this sample proportion is greater than 5%.
b) The result is not unusual because the probability that p^ is greater than or equal to this sample proportion is less than 5%.
c) The result is unusual because the probability that p^ is greater than or equal to this sample proportion is greater than 5%.
d) The result is unusual because the probability that p^ is greater than or equal to this sample proportion is less than 5%.
Explanation / Answer
a)
OPTION D: Approximately normal because n <= 0.05N and np(1-p) >= 10. [ANSWER]
Here, n = 300, p = 0.37, so np(1-p) = 93.24 > 10. ALso, this is surely less than 5% of all adults.
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b)
It is the population proportion,
u(p^) = p = 0.37 [ANSWER]
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c)
s = standard deviation = sqrt(p(1-p)/n) = 0.024140215 = 0.024 [ANSWER]
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D)
We first get the z score for the critical value. As z = (x - u) / s, then as
x = critical value = 0.34
u = mean = p = 0.37
s = standard deviation = sqrt(p(1-p)/n) = 0.024140215
Thus,
z = (x - u) / s = -1.242739532
Thus, using a table/technology, the left tailed area of this is
P(z < -1.242739532 ) = 0.106981916 [ANSWER]
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e)
We first get the z score for the critical value. As z = (x - u) / s, then as
x = critical value = 164/400 = 0.41
u = mean = p = 0.37
s = standard deviation = sqrt(p(1-p)/n) = 0.024140215
Thus,
z = (x - u) / s = 1.656986043
Thus, using a table/technology, the right tailed area of this is
P(z > 1.656986043 ) = 0.048761148
Hence,
OPTION D: d) The result is unusual because the probability that p^ is greater than or equal to this sample proportion is less than 5%. [ANSWER]
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