Work through the questions first, then open the answer key and explanations to check your work.
A transportation company randomly surveyed 80 employees who commute to work by train. The survey found that the mean commute time for the sample was 42 minutes, with a margin of error of 3 minutes. Which statement about the mean commute time for all employees at the company who commute by train is plausible?
A community garden has 1,200 members. The organizers randomly surveyed 60 members and found that 45 of them grow tomatoes in their plots. Based on the survey results, which choice is the best estimate of the number of members in the entire community garden who grow tomatoes?
A city library system has 240 registered teen members. A random sample of these members was surveyed, and 45% said they plan to attend the library's summer reading workshops. Based on this sample, what is the best estimate of the number of registered teen members who plan to attend the workshops?
A city planner randomly surveyed households in a town to estimate the mean monthly electricity usage for all households in the town. The survey estimated the mean to be 412.5 kilowatt-hours, with a margin of error of 8.5 kilowatt-hours.
Which choice is the best interpretation of this estimate?
A public transportation agency surveyed a random sample of city residents about the number of minutes they spend commuting to work each day. Based on the sample, the agency estimated that the mean commute time for all city residents is 38 minutes, with a margin of error of 4 minutes.
Which choice is the best interpretation of this estimate?
A city transportation department randomly surveyed residents about whether they use public buses at least once each week. Based on the survey results, the department estimated that 0.58 of all residents use public buses weekly, with a margin of error of 0.04.
Which choice is the most appropriate conclusion about the true proportion of all residents who use public buses at least once each week?
A community recreation department surveyed a random sample of 600 residents from a town with 40,000 residents. In the survey, 42% of residents said they would use a new indoor walking track if it were built. The survey results have a margin of error of 4 percentage points.
Based on the survey, which of the following is a plausible number of residents in the town who would use the new indoor walking track?
A community center surveyed a random sample of 480 adults who live in a town. In the survey, 62% of the adults said they use the town's walking trails at least once each month. The survey results have a margin of error of 4 percentage points.
Which choice is the most plausible conclusion about all adults who live in the town?
A community center surveyed a random sample of residents about which of two proposed weekend programs they preferred. The results are shown in the table.
| Program | Number of Residents |
|---|---|
| Outdoor Fitness Program | 348 |
| Arts Workshop Program | 252 |
If 9,000 residents are expected to participate in the final vote and the sample results are representative of all residents, by about how many votes is the Outdoor Fitness Program expected to win?
A city transportation department randomly surveyed 180 commuters and estimated that the mean one-way commute time for all commuters in the city is 34.5 minutes, with a margin of error of 2.1 minutes.
Which choice is the best interpretation of this estimate?
A city transportation department randomly surveyed 420 commuters and found that 58% of them use public transportation at least three times per week. The survey report states that the estimate has a margin of error of 4 percentage points.
Which choice is the most appropriate conclusion about the percentage of all commuters in the city who use public transportation at least three times per week?
The population mean commute time is between 39 and 45 minutes.
A margin of error of 3 minutes means the population mean is plausibly within 3 minutes of the sample mean of 42 minutes. Subtracting and adding 3 gives an interval from 39 to 45 minutes.
\(42-3=39\) and \(42+3=45\). Therefore, the plausible statement is that the population mean commute time is between 39 and 45 minutes.
900
In the random sample, \(\frac{45}{60}=\frac{3}{4}\) of the members grow tomatoes. Apply this proportion to the full population of 1,200 members:
\(\frac{3}{4}\times 1200=900\)
So the best estimate is 900 members.
108
To estimate the number of teen members who plan to attend, multiply the total number of members by the sample percentage written as a decimal:
\(240 \times 0.45 = 108\)
So the best estimate is 108 members. Choice C is correct.
Choice A uses the percentage itself instead of finding the estimated count. Choice B finds the number who do not plan to attend: \(240-108=132\). Choice D repeats the total population size.
The mean monthly electricity usage for all households in the town is likely between 404.0 and 421.0 kilowatt-hours.
A margin of error of 8.5 kilowatt-hours means the population mean is plausibly within 8.5 of the estimate 412.5. Subtracting gives \(412.5-8.5=404.0\), and adding gives \(412.5+8.5=421.0\). So the best conclusion is that the mean monthly electricity usage for all households in the town is likely between 404.0 and 421.0 kilowatt-hours.
It is reasonable that the true mean commute time for all city residents is between 34 and 42 minutes.
A margin of error of 4 minutes around an estimated mean of 38 minutes gives a plausible interval from 34 minutes to 42 minutes. The best conclusion is that the true mean commute time for all city residents is likely somewhere in that interval. The estimate does not guarantee the true mean is exactly 38 minutes, and it does not mean every individual resident has a commute time between 34 and 42 minutes. Also, values outside the interval are not impossible; they are just less supported by the sample estimate.
The true proportion is likely between 0.54 and 0.62.
The estimated proportion is 0.58 and the margin of error is 0.04. Subtracting and adding the margin of error gives a plausible interval for the true population proportion:
\(0.58-0.04=0.54\) and \(0.58+0.04=0.62\).
So the true proportion is reasonably likely to be between 0.54 and 0.62. Therefore, choice C is correct.
16,800
The estimated percentage is 42% with a margin of error of 4 percentage points, so the plausible percentage range is from 38% to 46%.
Now convert these percentages to numbers of residents in the full population of 40,000:
\(0.38(40{,}000)=15{,}200\)
\(0.46(40{,}000)=18{,}400\)
So a plausible number of residents who would use the walking track must be between 15,200 and 18,400. Of the choices given, only 16,800 falls within this interval.
Between 58% and 66% of all adults in the town likely use the walking trails at least once each month.
The survey estimate is 62%, and the margin of error is 4 percentage points. A plausible interval for the population percentage is found by subtracting and adding the margin of error:
\(62\%-4\%=58\%\) and \(62\%+4\%=66\%\).
So it is reasonable to conclude that between 58% and 66% of all adults in the town use the walking trails at least once each month. Therefore, choice C is correct.
1,440
The sample included \(348+252=600\) residents. The difference between the two programs in the sample was \(348-252=96\) votes.
The proportion representing the winning margin is \(\frac{96}{600}=0.16\).
If 9,000 residents vote, the expected winning margin is \(0.16\times 9000=1440\).
Therefore, the Outdoor Fitness Program is expected to win by about 1,440 votes.
The mean commute time for all commuters in the city is plausibly between 32.4 minutes and 36.6 minutes.
A margin of error of 2.1 minutes means plausible values for the population mean commute time are within 2.1 minutes of 34.5 minutes. Calculate the interval: \(34.5-2.1=32.4\) and \(34.5+2.1=36.6\). Therefore, the population mean commute time is plausibly between 32.4 minutes and 36.6 minutes.
It is plausible that between 54% and 62% of all commuters in the city use public transportation at least three times per week.
The survey estimate is 58%, and the margin of error is 4 percentage points. A plausible interval for the true population percentage is found by subtracting and adding the margin of error: \(58-4=54\) and \(58+4=62\). So it is plausible that between 54% and 62% of all commuters in the city use public transportation at least three times per week. Choice C correctly interprets the estimate using the margin of error and uses appropriate uncertainty language.
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