Confidence Interval for a Population Mean - AP Statistics

1. What happens to the interval if sample standard deviation decreases?

   The interval becomes narrower. Smaller $s$ reduces standard error and margin of error.

2. How does variability affect the confidence interval?

   Greater variability widens the interval. Higher $s$ increases standard error and interval width.

3. Find the 95% confidence interval: $\bar{x} = 10$, $t^* = 2$, $s = 5$, $n = 25$.

   $10 \pm 2$. Margin of error is $2 \times \frac{5}{\sqrt{25}} = 2 \times 1 = 2$.

4. Find $t^*$ for $df = 10$ at 95% confidence level.

   Approximately 2.228. From $t$-table with $df = 10$ and 95% confidence level.

5. Which option increases the precision of a confidence interval?

   Increasing sample size. Larger sample reduces standard error and interval width.

6. Find the degrees of freedom for $n = 15$.

   14. Always $n - 1$ for single-sample $t$-procedures.

7. What happens to a confidence interval if the confidence level decreases?

   The interval becomes narrower. Lower confidence level uses smaller critical value.

8. What does a 95% confidence level imply?

   95% of such intervals will capture the true population mean. Long-run frequency of intervals containing true parameter.

9. State the formula for a confidence interval for a mean.

   $\bar{x} \, \pm \, t^* \left(\frac{s}{\sqrt{n}}\right)$. Sample mean plus/minus margin of error using $t$-distribution.

10. Which table do you use to find $t^*$ values?

    The $t$-distribution table. Critical values depend on confidence level and degrees of freedom.

11. Calculate the standard error: $s = 3$, $n = 36$.

    $\frac{3}{\sqrt{36}} = 0.5$. Standard error formula: $s$ divided by square root of $n$.

12. What does $t^*$ represent in the confidence interval formula?

    Critical value from the $t$-distribution. Based on confidence level and degrees of freedom.

13. What is the relationship between confidence level and margin of error?

    Higher confidence increases margin of error. Higher confidence requires larger critical value and margin.

14. State the formula for standard error of the mean.

    $\frac{s}{\sqrt{n}}$. Standard deviation of the sampling distribution of $\bar{x}$.

15. What is the effect of confidence level on interval width?

    Higher confidence level widens the interval. More confidence requires wider interval to capture parameter.

16. Calculate confidence interval: $\bar{x} = 20$, $t^* = 1.96$, $s = 4$, $n = 100$.

    $20 \pm 0.784$. Margin of error: $1.96 \times \frac{4}{\sqrt{100}} = 1.96 \times 0.4 = 0.784$.

17. What does $\bar{x}$ represent in the confidence interval formula?

    Sample mean. The average of all observations in the sample.

18. How do you interpret a confidence interval?

    A range where the population mean is likely to be found. Plausible values for the unknown population mean.

19. Identify the condition for using a $t$-distribution in confidence intervals.

    Population standard deviation is unknown and sample is small. Use when $\sigma$ is unknown, especially with small samples.

20. How does sample size relate to the standard error?

    Larger sample size decreases standard error. Standard error decreases as $n$ increases: $\frac{s}{\sqrt{n}}$.

21. Calculate the interval: $\bar{x} = 50$, $t^* = 2.5$, $s = 10$, $n = 16$.

    $50 \pm 6.25$. Margin of error: $2.5 \times \frac{10}{\sqrt{16}} = 2.5 \times 2.5 = 6.25$.

22. Which term describes the range of values in a confidence interval?

    Margin of error. Half-width of confidence interval around point estimate.

23. Which distribution is used when population standard deviation is known?

    Normal distribution (Z-distribution). Known $\sigma$ allows use of standard normal distribution.

24. Find the margin of error given $t^* = 2$, $s = 4$, $n = 16$.

    $2 \times \frac{4}{\sqrt{16}} = 2$. Margin of error equals $t^* \times$ standard error.

25. What does $n$ represent in the confidence interval formula?

    Sample size. Total number of observations in the sample.

26. Identify the conditions for normality in constructing confidence intervals.

    Sample size large or population distribution approximately normal. Ensures sampling distribution of $\bar{x}$ is approximately normal.

27. Identify the effect of increasing sample size on the confidence interval.

    It narrows the confidence interval. Larger $n$ reduces standard error and margin of error.

28. What is the central limit theorem in context of confidence intervals?

    The sampling distribution of the sample mean is approximately normal. Allows use of normal approximation for sample means.

29. Identify the effect of decreasing variability on the confidence interval.

    It narrows the interval. Less variability reduces standard error and margin of error.

30. Which factor has no effect on interval width if other factors are fixed?

    Sample mean. Sample mean is center point; doesn't affect interval width.

31. Identify the sample mean in the data set: 5, 6, 7, 8, 9.

    $\bar{x} = 7$. Sum of values divided by count: $(5+6+7+8+9)/5 = 7$.

32. What does a wider confidence interval suggest about precision?

    Less precision. Wider intervals indicate greater uncertainty about parameter.

33. What does $s$ represent in the confidence interval formula?

    Sample standard deviation. Measures spread of sample observations around sample mean.

34. Identify the degrees of freedom for a sample size of 25.

    24. Degrees of freedom equals $n - 1 = 25 - 1 = 24$.

35. What is the primary purpose of a confidence interval?

    To estimate a population parameter. Uses sample data to infer about population parameters.

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