Question

Math

Posted 6 months ago

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A large company is considering installing charging stations for electric cars in their company parking lots. Suppose that $1.5 \%$ of employees at the main office have electric cars and $2 \%$ of employees at the branch office have electric cars.
The company wonders whether there is greater need for the stations at the main office. They plan to take separate random samples of 100 employees from the main office and 50 employees from a branch office. Then they will look at the difference in the sample proportions $\left(\hat{p}_{M}-\hat{p}_{\mathrm{B}}\right)$.
What will be the shape of the sampling distribution of $\hat{p}_{M}-\hat{p}_{B}$, and why?
Choose 1 answer:
(A) Approximately normal, because we expect 11 electric and 139 non-electric cars at each office, and all of these counts are at least 10.
(B) Approximately normal, because we expect 15 electric and 85 nonelectric cars at the main office, and 10 and 40 at the branch office, and all of these counts are at least 10 .
(C) Not normal, because we expect fewer than 10 electric cars in both samples.
(D) Not normal, because we expect fewer than 10 non-electric cars in both samples.
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Answer from Sia

Posted 6 months ago

Solution by Steps

step 1

Calculate the expected number of electric cars in the main office sample

step 2

For the main office: $E_{M} = n_{M} \times p_{M} = 100 \times 0.015 = 1.5$ electric cars

step 3

Calculate the expected number of electric cars in the branch office sample

step 4

For the branch office: $E_{B} = n_{B} \times p_{B} = 50 \times 0.02 = 1$ electric car

step 5

Determine if the expected counts meet the criteria for normal approximation

step 6

Since $E_{M} < 10$ and $E_{B} < 10$, the expected counts of electric cars do not meet the criteria for normal approximation

step 7

Conclude the shape of the sampling distribution based on the expected counts

step 8

The sampling distribution of $\hat{p}_{M}-\hat{p}_{B}$ will not be approximately normal because the expected number of electric cars in both samples is less than 10

[question number] Answer

The correct answer is (C) Not normal, because we expect fewer than 10 electric cars in both samples.

Key Concept

Normal Approximation Criteria for Proportions

Explanation

The sampling distribution of the difference in sample proportions can be approximated by a normal distribution if the expected number of successes and failures in both samples are at least 10. In this case, the expected number of electric cars (successes) in both samples is less than 10, so the distribution will not be normal.

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