Justifying Claims: Population Mean, Confidence Interval - AP Statistics

1. What affects the width of a confidence interval for a mean?

   Sample size and variability. Larger $n$ reduces width; higher variability increases width.

2. What happens to the CI width if the confidence level is increased?

   The width increases. Higher confidence requires larger critical value, increasing margin of error.

3. What distribution is used for large sample sizes in CI estimation?

   Normal distribution. Central Limit Theorem ensures sampling distribution is approximately normal.

4. What does $z^*$ represent in the confidence interval formula?

   The critical value from the standard normal distribution. From standard normal table, depends on confidence level chosen.

5. What is the purpose of a confidence interval?

   To estimate a population parameter. Provides range of plausible values for unknown population parameter.

6. How does sample size affect the reliability of a CI?

   Larger samples increase reliability. Larger samples reduce sampling variability and increase precision.

7. What is the effect of a higher confidence level on the CI?

   Wider interval. Larger critical value needed increases the margin of error.

8. What happens to the CI width if the sample size increases?

   The width decreases. Standard error decreases as $n$ increases, reducing margin of error.

9. What is the effect of a lower confidence level on the CI?

   Narrower interval. Smaller critical value decreases the margin of error.

10. What does a wider confidence interval imply about precision?

    Less precision. Wider intervals indicate greater uncertainty in the estimate.

11. When is the $t$-distribution used instead of the normal distribution?

    When population standard deviation is unknown and $n < 30$. Accounts for additional uncertainty when $\sigma$ is unknown with small samples.

12. What is the impact of a larger sample size on margin of error?

    Reduces margin of error. Standard error decreases with $\sqrt{n}$, reducing uncertainty.

13. What is the standard error in the context of a CI for mean?

    $\frac{s}{\text{√}n}$. Measures variability of sample mean as estimate of population mean.

14. If a 95% CI is (10, 20), what is the point estimate?

    15. Midpoint of interval endpoints: $(10 + 20)/2 = 15$.

15. What is $\bar{x}$ in the context of a confidence interval?

    Sample mean. Point estimate for the population mean from the sample data.

16. Which confidence interval is wider, 90% or 95%?

    95%. Higher confidence level requires wider interval to maintain reliability.

17. What is the formula for a confidence interval for a population mean?

    $\bar{x} \text{ ± } z^* \frac{s}{\text{√}n}$. Standard formula: sample mean plus/minus critical value times standard error.

18. For a 99% CI, what is the approximate $z^*$ value?

    2.576. Critical value leaving 0.5% in each tail of standard normal distribution.

19. What happens to the standard error if $n$ is quadrupled?

    It halves. Standard error is inversely proportional to $\sqrt{n}$.

20. Which condition must be checked before using a $t$-interval?

    Data must be approximately normal if $n < 30$. Small samples require normality assumption for valid inference.

21. Identify the margin of error in the confidence interval formula.

    $z^* \frac{s}{\text{√}n}$. Measures uncertainty in the estimate based on sample variability.

22. For a 90% CI, what is the approximate $z^*$ value?

    1.645. Critical value leaving 5% in each tail of standard normal distribution.

23. Which sample size gives a more precise estimate, $n=30$ or $n=100$?

    $n=100$. Larger sample size reduces standard error, giving narrower interval.

24. If the sample mean is 50 and margin of error is 5, find CI range.

    45 to 55. Sample mean ± margin of error gives the interval bounds.

25. What is the first step in constructing a confidence interval?

    Calculate the sample mean, $\bar{x}$. Point estimate serves as center of the confidence interval.

26. What is the appropriate $t^*$ critical value for a 95% CI with df=9?

    Approximately 2.262. From $t$-table with $df = 9$ and $\alpha/2 = 0.025$.

27. Identify the population parameter estimated by a confidence interval.

    Population mean. The unknown parameter the interval is designed to estimate.

28. How does increasing the sample variability affect CI width?

    Increases the width. Higher $s$ increases standard error and thus margin of error.

29. When is a confidence interval considered valid?

    When assumptions about sample and distribution hold. Requires proper sampling method and satisfied distributional assumptions.

30. What is the implication of a confidence interval that includes zero?

    Possible lack of effect or no difference. Zero in interval suggests no significant difference from zero.

31. What does a 95% CI of (30, 40) imply about the population mean?

    We are 95% confident it lies between 30 and 40. Interval interpretation: plausible range for unknown population parameter.

32. Why is a confidence interval not a probability statement?

    It describes a range, not a probability of a single event. Interval captures parameter or not; confidence refers to method reliability.

33. What calculation provides the midpoint of a confidence interval?

    Sample mean, $\bar{x}$. Center of symmetric interval equals the point estimate.

34. Identify the critical value used for 95% confidence interval.

    $z^* = 1.96$. Standard value for 95% confidence from normal distribution table.

35. What does $s$ represent in the CI formula?

    Sample standard deviation. Estimates population standard deviation from sample data.

36. What are the degrees of freedom for a $t$-distribution with $n=15$?

    14. Degrees of freedom equals $n - 1$ for one-sample $t$-procedures.

37. What condition must be met for a confidence interval based on a normal distribution?

    Population distribution must be normal or sample size large. Central Limit Theorem applies when $n \geq 30$ for non-normal populations.

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