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STA258 Lecture 08
STA258 Lecture 08 Raw
STA258 Lecture 08 Flashcards
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Completed Notes Status
- Completed insertions: 4 (Cleaned up formatting, fixed typos in "set of", clarified S vs S^2 ambiguity in ancient air example)
- Ambiguities left unresolved: 1 (Data vs Summary statistics mismatch in Ancient Air example; used provided summary stats)
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Lecture Summary
- Central objective: Derive and apply confidence intervals for population means (using Z and T distributions), proportions, and variances, and determine necessary sample sizes for desired precision.
- Key concepts:
- Z-Interval for Mean: Used when the population variance (
) is known or is large. Based on the standard normal distribution ( ). - T-Interval for Mean: Used when variance is unknown and estimated by
, typically for smaller samples ( ). Relies on the -distribution with degrees of freedom. - Sample Size Determination: Calculating the minimum
required to bound the margin of error ( ) within a specific confidence level ( ), for both means and proportions. - Confidence Interval for Variance: Uses the Chi-square (
) distribution pivots to estimate the population variance or standard deviation.
- Z-Interval for Mean: Used when the population variance (
- Connections:
- The Central Limit Theorem justifies the use of normal approximations for large samples (
) even when the underlying population is not normal. - The 10% Condition allows for the assumption of independence in sampling without replacement if the sample is small relative to the population.
- The Central Limit Theorem justifies the use of normal approximations for large samples (
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TK Resolutions
- #tk: Large sample Confidence Intervals are on the test.
- Answer: Confirmed scope for Test 1 includes Z-intervals for means and proportions.
- #tk: Can we optimize the tradeoff? Get the most confidence with the least interval.
- Answer: This is addressed by Sample Size Determination. By fixing the margin of error (
) and confidence level ( ) in advance, we solve for . Increasing narrows the interval width without sacrificing confidence.
- Answer: This is addressed by Sample Size Determination. By fixing the margin of error (
- #tk: Learn Q-Q plots Normal Q-Q Plot.
- Answer: Q-Q plots are critical for verifying the normality assumption required for small-sample T-intervals. (See atomic note: Normal Q-Q Plot).
- #tk: Large sample Confidence Intervals are on the test.
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Practice Questions
- Remember/Understand:
- What is the critical difference between choosing a Z-statistic and a T-statistic for a mean confidence interval?
- Why is the confidence interval for variance not symmetric around the point estimate?
- What value should be used for
when calculating sample size if no prior estimate is available?
- Apply/Analyze:
- A pilot study gives a standard deviation of 15. Calculate the sample size needed to estimate the mean within 2 units at 95% confidence.
- You compute a 95% CI for a mean as (10, 20). If you increase the confidence level to 99% without changing
, what happens to the interval width?
- Evaluate/Create:
- Evaluate the validity of a T-interval calculation performed on a sample of size
with strong skewness evident in the histogram. - Propose a sampling plan for a political poll that ensures a margin of error of no more than 3% without knowing the current candidate popularity.
- Evaluate the validity of a T-interval calculation performed on a sample of size
- Remember/Understand:
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Challenging Concepts
- Confidence Interval for Variance:
- Why it's challenging: Unlike Z or T intervals, the Chi-square distribution is not symmetric, meaning the distance from the point estimate to the lower bound differs from the distance to the upper bound.
- Study strategy: Practice drawing the non-symmetric Chi-square curve and explicitly labeling the areas
and to ensure the correct critical values are placed in the denominator.
- Sample Size Determination:
- Why it's challenging: Requires algebraic manipulation of the interval formula before data collection, and understanding the "worst-case" scenario for proportions (
). - Study strategy: Memorize the isolated
formulas and the logic behind maximizing .
- Why it's challenging: Requires algebraic manipulation of the interval formula before data collection, and understanding the "worst-case" scenario for proportions (
- Confidence Interval for Variance:
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Action Plan
- Immediate review actions:
- Practice and application:
- Deep dive study:
- Verification and integration:
- Immediate review actions:
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Footnotes