SAT Inference & Margin of Error Question 27 | Free SAT Prep | Aniko

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Question 27·Hard·Inference from Sample Statistics and Margin of Error

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Rent Survey Results

Statistic	Value
Sample size	400 apartments
Sample mean monthly rent	$1,120
Margin of error (95% confidence)	± $60

There are 18,500 apartments in the city. Based on the survey results, what is the greatest possible total amount of rent that all apartments in the city could collect in a single month, in dollars?

For questions involving a sample mean and a margin of error, first turn the margin of error into an interval by adding and subtracting from the sample mean. Then decide whether the problem wants the minimum, maximum, or just a typical value, and choose the lower bound, upper bound, or center accordingly. Finally, apply that value to the full population by multiplying by the total number of items (here, apartments), not the sample size. Doing the interval step and then a single clear multiplication keeps the work quick and avoids common traps.

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Use the margin of error correctly

The margin of error is $\pm 60$ dollars around the sample mean of $1{,}120$ . What is the highest plausible value for the true average monthly rent?

Focus on "greatest possible"

Since the question wants the greatest possible total rent, should you use the lower bound, the upper bound, or the middle (sample mean) of the interval?

Connect average rent to total rent

Once you know the largest possible average rent per apartment, how can you use the total number of apartments, $18{,}500$ , to find the total rent collected in one month?

Check which quantity to multiply

Be careful to multiply by the total number of apartments in the city (18,500), not the sample size (400).

Desmos Guide

Compute the maximum total rent

In Desmos, type the expression (1120 + 60) * 18500 to represent the highest possible average rent ( $1{,}120 + 60$ ) times the 18,500 apartments. The number Desmos outputs is the greatest possible total monthly rent in dollars.

Step-by-step Explanation

Interpret the margin of error

The sample mean monthly rent is $1{,}120$ , with a margin of error of $\pm 60$ dollars at a 95% confidence level.

That means the true average monthly rent for all apartments in the city is likely between:

Lower bound: $1{,}120 - 60 = 1{,}060$
Upper bound: $1{,}120 + 60 = 1{,}180$

Decide which bound to use

The question asks for the greatest possible total amount of rent all apartments could collect.

To make the total as large as possible, we want the largest possible average rent, which is the upper bound of the interval: $1{,}180$ dollars per month.

Set up the total rent calculation

If the true average rent for the city were $1{,}180$ dollars, then the total monthly rent collected by all $18{,}500$ apartments would be:

\text{Total rent} = 18{,}500 \times 1{,}180

This expression represents the greatest possible total monthly rent consistent with the survey and margin of error.

Compute the total rent

Now multiply $18{,}500$ by $1{,}180$ :

\begin{aligned} 18{,}500 \times 1{,}180 &= (18{,}000 + 500) \times 1{,}180 \\ &= 18{,}000 \times 1{,}180 + 500 \times 1{,}180 \\ &= 21{,}240{,}000 + 590{,}000 \\ &= 21{,}830{,}000 \end{aligned}

So the greatest possible total monthly rent is $21{,}830{,}000$ dollars, which corresponds to answer choice C) 21,830,000.

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Desmos Guide

Step-by-step Explanation

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Desmos Guide

Step-by-step Explanation