Explain confidence intervals.
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What is a Confidence Interval?
A confidence interval (CI) is a range of values, calculated from sample data, that is likely to contain the true population parameter (like the mean or proportion) with a certain level of confidence.
For example:
If you survey 100 people and estimate the average height is 170 cm with a 95% confidence interval of 167–173 cm, it means you are 95% confident that the true average height of the entire population lies between 167 and 173 cm.
Key Ideas
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Point Estimate vs. Interval Estimate
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A point estimate (e.g., sample mean = 170 cm) gives a single number.
-
A confidence interval gives a range, acknowledging uncertainty in the estimate.
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Confidence Level
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Common levels: 90%, 95%, 99%.
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A 95% CI means: if we repeated the sampling process many times, about 95% of the intervals would contain the true population value.
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Width of the Interval
-
Narrow CI → more precise estimate (usually from large samples).
-
Wide CI → less precision (often from small samples or high variability).
-
Factors Affecting CI
-
Sample Size: Larger samples → narrower intervals.
-
Variability: More spread in data → wider intervals.
-
Confidence Level: Higher confidence (like 99%) → wider interval, since you want more certainty.
Point Estimate vs. Interval Estimate
-
A point estimate (e.g., sample mean = 170 cm) gives a single number.
-
A confidence interval gives a range, acknowledging uncertainty in the estimate.
Confidence Level
-
Common levels: 90%, 95%, 99%.
-
A 95% CI means: if we repeated the sampling process many times, about 95% of the intervals would contain the true population value.
Width of the Interval
-
Narrow CI → more precise estimate (usually from large samples).
-
Wide CI → less precision (often from small samples or high variability).
Factors Affecting CI
-
Sample Size: Larger samples → narrower intervals.
-
Variability: More spread in data → wider intervals.
-
Confidence Level: Higher confidence (like 99%) → wider interval, since you want more certainty.
✅ In short:
A confidence interval gives not just an estimate, but also the margin of error around it, showing how reliable the estimate is. It tells you the range where the true population parameter is likely to fall, with a chosen level of confidence.
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