Inference from Sample Statistics and Margin of Error Flashcards

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?

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.

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?

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).

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?

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.

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?

Answer: 40%

The 95% confidence interval is 40.5%–47.5%; 40% is below the lower bound and therefore not in the interval.

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?

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.

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?

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.

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?

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.