Confidence Intervals and Estimation Flashcards
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A 95% confidence interval for the mean hours of sleep per night is (6.8, 7.4). A classmate says 'the true mean is definitely in this interval.' This claim is:
Answer: Incorrect; we are 95% confident the interval captures the true mean, but it might not.
No single confidence interval is guaranteed to contain the true mean; the 95% refers to the long-run success rate of the procedure.
A two-sample t-interval for μ1 − μ2 does not include 0. This suggests:
Answer: There is convincing evidence that the two population means differ.
When 0 is not in the confidence interval for a difference, it provides convincing evidence (at that confidence level) that the means differ.
Which of the following is a point estimate, not an interval estimate?
Answer: x̄ = 42.6
A point estimate is a single value (like x̄) used to estimate the parameter, while an interval estimate gives a range.
A confidence interval is said to be 'more precise' when:
Answer: The interval is narrower.
Precision in estimation refers to the width of the interval — a narrower interval provides a more precise estimate of the parameter.
When computing a confidence interval for a population proportion, which value is used for p in the standard error formula?
Answer: p̂, the sample proportion
Since p is unknown, we substitute p̂ (the sample proportion) when computing the standard error for a confidence interval.
A 90% CI for a mean is (14.2, 19.8) and a 99% CI for the same data is (12.1, 21.9). Which interval is more likely to contain the true population mean?
Answer: The 99% CI, because it was constructed with a higher confidence level.
A 99% CI captures the true mean in 99% of samples in the long run, compared to 90% for the narrower interval.
For a paired t-interval, data are collected as differences (d = x1 − x2) for each pair. The interval is then constructed as:
Answer: d̄ ± t* × (sd/√n)
A paired t-interval treats the differences as a single sample, using d̄ and sd, with t* based on n − 1 degrees of freedom.