Question

In 2016 the Better Business Bureau settled 80% of complaints they received in the United States. Suppose you have been hired by the Better Business Bureau to investigate the complaints they received this year involving new car dealers. You plan to select a sample of new car dealer complaints to estimate the proportion of complaints the Better Business Bureau is able to settle. Assume the population proportion of complaints settled for new car dealers is 0.80, the same as the overall proportion of complaints settled in 2016.

(a)

Suppose you select a sample of 180 complaints involving new car dealers. Show the sampling distribution of

p.

A bell-shaped curve is above a horizontal axis labeled p.

• In order of left to right, the ticks on the horizontal axis are labeled: 0.74, 0.77, 0.8, 0.83, 0.86.
• The curve enters the viewing window near 0.74 just above the horizontal axis and travels up to the right to a maximum near 0.8.
• After reaching the maximum, the curve then travels down and to the right until it leaves the viewing window at the same height it entered near 0.86.

A bell-shaped curve is above a horizontal axis labeled p.

• In order of left to right, the ticks on the horizontal axis are labeled: −0.06, −0.03, 0, 0.03, 0.06.
• The curve enters the viewing window near −0.06 just above the horizontal axis and travels up to the right to a maximum near 0.
• After reaching the maximum, the curve then travels down and to the right until it leaves the viewing window at the same height it entered near 0.06.

A bell-shaped curve is above a horizontal axis labeled p.

• In order of left to right, the ticks on the horizontal axis are labeled: −1.2, −0.2, 0.8, 1.8, 2.8.
• The curve enters the viewing window near −1.2 just above the horizontal axis and travels up to the right to a maximum near 0.8.
• After reaching the maximum, the curve then travels down and to the right until it leaves the viewing window at the same height it entered near 2.8.

A bell-shaped curve is above a horizontal axis labeled p.

• In order of left to right, the ticks on the horizontal axis are labeled: −0.03, 0, 0.03, 0.06, 0.09.
• The curve enters the viewing window near −0.03 just above the horizontal axis and travels up to the right to a maximum near 0.03.
• After reaching the maximum, the curve then travels down and to the right until it leaves the viewing window at the same height it entered near 0.09.

(b)

Based upon a sample of 180 complaints, what is the probability that the sample proportion will be within 0.04 of the population proportion? (Round your answer to four decimal places.)

(c)

Suppose you select a sample of 470 complaints involving new car dealers. Show the sampling distribution of

p.

A bell-shaped curve is above a horizontal axis labeled p.

• In order of left to right, the ticks on the horizontal axis are labeled: −0.04, −0.02, 0, 0.02, 0.04.
• The curve enters the viewing window near −0.04 just above the horizontal axis and travels up to the right to a maximum near 0.
• After reaching the maximum, the curve then travels down and to the right until it leaves the viewing window at the same height it entered near 0.04.

A bell-shaped curve is above a horizontal axis labeled p.

• In order of left to right, the ticks on the horizontal axis are labeled: −1.2, −0.2, 0.8, 1.8, 2.8.
• The curve enters the viewing window near −1.2 just above the horizontal axis and travels up to the right to a maximum near 0.8.
• After reaching the maximum, the curve then travels down and to the right until it leaves the viewing window at the same height it entered near 2.8.

A bell-shaped curve is above a horizontal axis labeled p.

• In order of left to right, the ticks on the horizontal axis are labeled: 0.76, 0.78, 0.8, 0.82, 0.84.
• The curve enters the viewing window near 0.76 just above the horizontal axis and travels up to the right to a maximum near 0.8.
• After reaching the maximum, the curve then travels down and to the right until it leaves the viewing window at the same height it entered near 0.84.

A bell-shaped curve is above a horizontal axis labeled p.

• In order of left to right, the ticks on the horizontal axis are labeled: −0.02, 0, 0.02, 0.04, 0.06.
• The curve enters the viewing window near −0.02 just above the horizontal axis and travels up to the right to a maximum near 0.02.
• After reaching the maximum, the curve then travels down and to the right until it leaves the viewing window at the same height it entered near 0.06.

(d)

Based upon the larger sample of 470 complaints, what is the probability that the sample proportion will be within 0.04 of the population proportion? (Round your answer to four decimal places.)

(e)

As measured by the increase in probability, how much do you gain in precision by taking the larger sample in part (d)?

Answer 1

Based on the answering guidelines, I'm answering the first four sub-parts.

(a) The bell-shaped curve is restricted from 0.74 to 0.86. As explained in Q, the population proportion of complaints settled for new car dealers is 0.8. Hence, in the bell-curved shape, the maxima is at 0.80, and the curve starts from 0.74 and ends at 0.86. The plot looks like this where the x-axis represents the proportion of complaints settled for new car dealers, and the y-axis represents the respective probabilities. In this case, it can be assumed that takes the value 1 in case the ith complaint is settled, and takes the value 0 in case the complaint is not settled.

The above mentioned is a Bernoulli distribution having a mean of p and variance of p(1 - p). Given there are 180 data points, the mean can be represented as a normal distribution assuming iid for the complaints.

We can write: -

Hence,

(b) In case represents the sample proportion, we've using Central Limit Theorem: -

On solving,

On simplifying,

We can further write this: -

(c) In case the sample size is increased to 470 car dealers,

Hence,

(d) In case represents the sample proportion, we've using Central Limit Theorem: -

On solving,

On simplifying,

We can further write this: -

(e) Gain in precision can be calculated as 18.225%: -