Making Inferences and Justifying Conclusions: Estimating Population Parameters with Margin of Error (CCSS.S-IC.4)
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Common Core High School Statistics And Probability › Making Inferences and Justifying Conclusions: Estimating Population Parameters with Margin of Error (CCSS.S-IC.4)
A survey of 850 voters found that 52% support a new policy. A 90% confidence interval for the population proportion is (0.49, 0.55). Which interpretation of the confidence interval is correct?
Ninety percent of voters support the policy.
We are 90% confident that the true proportion of voters who support the policy is between 0.49 and 0.55.
There is a 90% chance that the true proportion is exactly 0.52.
Ninety percent of future polls will report support between 0.49 and 0.55.
Explanation
The interval estimates the population proportion; 90% confidence describes the long-run success rate of the method. It is not a statement about individuals, an exact probability for the parameter, or future samples.
Students recorded 36 morning bus rides on a route. The sample mean time was 18.4 minutes, and a 95% confidence interval for the population mean is (17.2, 19.6) minutes. Which interpretation of the confidence interval is correct?
Ninety-five percent of future bus rides will last between 17.2 and 19.6 minutes.
The probability that the sample mean is between 17.2 and 19.6 minutes is 95%.
The interval proves the district average is exactly 18.4 minutes.
We are 95% confident that the true mean bus ride time for all rides on this route is between 17.2 and 19.6 minutes.
Explanation
The correct interpretation focuses on the population mean $\mu$: we are 95% confident $\mu$ is between 17.2 and 19.6 minutes. The other statements describe individual rides, misstate probability about the sample mean, or treat the sample mean as the population mean.
A random sample of 150 teens reported an average of 6.8 hours of sleep per night. A 95% confidence interval for the population mean is (6.5, 7.1) hours. Which interpretation of the confidence interval is correct?
Ninety-five percent of teens sleep between 6.5 and 7.1 hours per night.
There is a 95% chance that the sample mean 6.8 hours lies in (6.5, 7.1).
We can be sure the true mean equals 6.8 hours because it is inside the interval.
We are 95% confident that the true average sleep for all teens is between 6.5 and 7.1 hours.
Explanation
Confidence intervals give a plausible range for the population mean; 95% confidence refers to the method's long-run capture rate. It does not describe individual teens, the probability of the sample mean, or guarantee the parameter equals the sample mean.
A city surveyed 40 households about daily water use. The sample mean was 180 gallons, and a 95% confidence interval for the population mean is (170, 190) gallons. Which interpretation of the confidence interval is correct?
The probability that $\mu$ is between 170 and 190 is 0.95.
We are 95% confident that the true mean daily water use per household in the city is between 170 and 190 gallons.
Ninety-five percent of households use between 170 and 190 gallons per day.
The sample mean will always be between 170 and 190 gallons.
Explanation
The interval is meant to capture the population mean $\mu$. We are 95% confident that $\mu$ is between 170 and 190 gallons. The other statements either assign probability to $\mu$ after observing data, describe individual household values instead of the mean, or confuse the sample mean with the interval.
A survey of 120 households found that 48 own a dog. A 95% confidence interval for the population proportion of households owning a dog is (0.31, 0.49). Which interpretation of the confidence interval is correct?
The probability that $p$ equals 0.40 is 95%.
We are 95% confident that the true proportion of households in the town that own a dog is between 0.31 and 0.49.
Ninety-five percent of the 120 households in the sample own a dog.
If we took many samples, 95% of the samples would have a dog-ownership rate between 0.31 and 0.49.
Explanation
A confidence interval provides plausible values for the population proportion $p$. We are 95% confident that $p$ is between 0.31 and 0.49. The other choices either assign probability to a fixed $p$, misstate a fact about the sample, or incorrectly claim most sample proportions will lie in this one interval.
In a survey of 500 registered voters, 61% supported a ballot measure. A 95% confidence interval for the population proportion is (0.57, 0.65). Which interpretation of the confidence interval is correct?
There is a 61% chance the measure will pass.
Ninety-five percent of voters support the measure.
We are 95% confident that the true proportion of registered voters who support the measure is between 0.57 and 0.65.
Ninety-five percent of samples will have a sample proportion of exactly 0.61.
Explanation
A confidence interval describes plausible values for the population proportion $p$. We are 95% confident that $p$ is between 0.57 and 0.65. The other options make irrelevant claims about passing, confuse the interval with a percentage of individuals, or claim all samples will yield the same sample proportion.
A random sample of 200 residents reported an average of 3.1 hours of screen time per day. A 95% confidence interval for the population mean is (2.7, 3.5) hours. Which interpretation of the confidence interval is correct?
There is a 95% probability that the true mean is between 2.7 and 3.5 hours.
Ninety-five percent of residents have daily screen times between 2.7 and 3.5 hours.
We are 95% confident that the true average screen time for all residents is between 2.7 and 3.5 hours.
The sample mean 3.1 hours falls within this interval with 95% probability.
Explanation
A confidence interval describes plausible values for the population mean; 95% confidence refers to the long-run method, not a probability about this specific interval, individual residents, or the sample mean.
A random sample of 220 students was surveyed about a new school lunch menu. The sample proportion who prefer the new menu was 0.60, and a 95% confidence interval for the population proportion is (0.54, 0.66). Which interpretation of the confidence interval is correct?
We are 95% confident that the true proportion of all students at the school who prefer the new menu is between 0.54 and 0.66.
There is a 95% chance that the sample proportion is between 0.54 and 0.66.
Exactly 95% of students in the population prefer the new menu.
Ninety-five percent of future samples will have a sample proportion between 0.54 and 0.66.
Explanation
A confidence interval estimates a population parameter. The correct interpretation is that we are 95% confident the true population proportion $p$ lies between 0.54 and 0.66. The other options misinterpret the interval as about the sample, assign probability to $p$ after observing data, or describe where most sample proportions will fall.
A random sample of 120 students found that 58% prefer online homework. A 95% confidence interval for the population proportion is (0.50, 0.66). Which interpretation of the confidence interval is correct?
We are 95% confident that the true proportion of all students who prefer online homework is between 0.50 and 0.66.
There is a 95% chance that the sample proportion 0.58 is between 0.50 and 0.66.
Ninety-five percent of students prefer online homework.
Ninety-five percent of future samples will have proportions between 0.50 and 0.66.
Explanation
A confidence interval gives a range of plausible values for the population parameter; here, we are 95% confident the true proportion lies between 0.50 and 0.66. It does not describe the sample proportion, individual students, or the behavior of future samples.
In a random sample of 400 commuters, 36% used public transit at least 3 days per week. A 95% confidence interval for the population proportion is (0.31, 0.41). Which interpretation of the confidence interval is correct?
We are 95% confident that the true proportion of commuters who use public transit at least 3 days a week is between 0.31 and 0.41.
There is a 95% chance that 36% is between 0.31 and 0.41.
Most commuters individually use transit between 31% and 41% of the time.
In 95% of future samples, the sample proportion will be exactly 0.36.
Explanation
The interval estimates the population proportion and the confidence level refers to the long-run performance of the method. It is not a probability statement about the observed sample statistic, about individual behavior, or about exact future sample outcomes.