Bluebook SAT Test Inference from Sample Statistics and Margin of Error 4 — Questions and Answers
Question 1: A researcher conducts a study and finds a 90% confidence interval for the mean is (55, 65). If the researcher recalculates using a 95% confidence level with the same data, what happens to the interval?
- The interval becomes narrower.
- The interval becomes wider. (Correct answer)
- The interval stays the same.
- The interval shifts to the right.
Correct answer: The interval becomes wider.
Higher confidence levels require wider intervals to capture the true parameter with greater certainty, so a 95% interval is wider than a 90% interval for the same data.
Question 2: A random sample of 225 students finds a mean GPA of 3.1 with a standard error of 0.04. What is the 95% confidence interval for the mean GPA, given that the critical value is approximately 2?
- (3.02, 3.18) (Correct answer)
- (3.06, 3.14)
- (3.00, 3.20)
- (2.90, 3.30)
Correct answer: (3.02, 3.18)
The 95% confidence interval is 3.1 ± 2(0.04) = 3.1 ± 0.08, giving the range (3.02, 3.18).
Question 3: A market research firm surveys 500 randomly selected shoppers. The survey estimates that 72% prefer Brand A, with a margin of error of ±4%. A competitor claims that fewer than 68% of shoppers prefer Brand A. Which statement best evaluates this claim?
- The claim is plausible because 68% is within the confidence interval. (Correct answer)
- The claim is false because the sample proportion is 72%.
- The claim is plausible because 68% equals the lower bound.
- The claim is not plausible because 68% falls below the lower bound of 68%.
Correct answer: The claim is plausible because 68% is within the confidence interval.
The confidence interval is 68%–76%; since 68% is the lower bound of the interval, it is a plausible population value, making the competitor's claim possible.
Question 4: In a poll of 800 randomly selected adults, 44% said they regularly recycle, with a margin of error of ±3.5 percentage points. Which value is NOT in the 95% confidence interval?
- 41%
- 47%
- 40% (Correct answer)
- 44%
Correct answer: 40%
The 95% confidence interval is 40.5%–47.5%; 40% is below the lower bound and therefore not in the interval.
Question 5: A biologist samples 36 rabbits from a large population and finds a mean ear length of 8.4 cm. She increases her sample to 144 rabbits, finding essentially the same mean. Which statement about the two estimates is true?
- The second sample has a larger margin of error because more data introduces more uncertainty.
- Both estimates have the same margin of error since the means are similar.
- The second sample has a smaller margin of error because the sample size is four times larger. (Correct answer)
- The first sample is more reliable because smaller samples are more tightly controlled.
Correct answer: The second sample has a smaller margin of error because the sample size is four times larger.
Quadrupling the sample size (36 to 144) halves the margin of error since margin of error ∝ 1/√n.
Question 6: A study reports a 95% confidence interval for mean sleep duration of (6.8 hours, 7.6 hours). A health agency claims adults should sleep at least 7.5 hours on average. Which statement is best supported by the interval?
- The agency's recommendation is not plausible given the data.
- The agency's recommendation of 7.5 hours is a plausible population mean. (Correct answer)
- The true mean sleep duration is exactly 7.2 hours.
- Adults are sleeping too little because the lower bound is 6.8 hours.
Correct answer: The agency's recommendation of 7.5 hours is a plausible population mean.
7.5 hours falls within the confidence interval (6.8, 7.6), making it a plausible value for the true population mean.
Question 7: A school's random survey of 100 students estimates that the mean time spent on homework is 2.4 hours with a margin of error of ±0.3 hours at 95% confidence. What additional information is needed to determine whether the true mean exceeds 2.5 hours?
- The exact standard deviation of the sample
- Whether 2.5 falls within or outside the confidence interval (Correct answer)
- The number of teachers who assign homework
- The median homework time, not just the mean
Correct answer: Whether 2.5 falls within or outside the confidence interval
To determine if 2.5 hours is plausible, you check whether it lies within the confidence interval (2.1, 2.7); since it does, the data cannot rule out a true mean above 2.5.
Bluebook SAT Test Inference from Sample Statistics and Margin of Error 4
A researcher conducts a study and finds a 90% confidence interval for the mean is (55, 65).
If the researcher recalculates using a 95% confidence level with the same data, what happens to the interval?