Statistics / Unit 8: Sampling Distributions and Confidence Intervals
8.6 What Affects the Width of an Interval
Notes
Margin of error $= z^* \cdot \text{SE}$, so anything that changes $z^*$ or SE changes the width.
- Higher confidence level gives a larger $z^*$ and a wider interval.
- Larger sample size gives a smaller SE and a narrower interval.
- To cut the margin of error in half, multiply the sample size by 4.
- If a value (such as 0.5) is inside the interval, it is a plausible value for the parameter.
- Example: $90\%$, $95\%$ and $99\%$ intervals from the same data get wider in that order.
Practice
A 90%, a 95% and a 99% confidence interval are made from one sample. Which interval is the narrowest?
Lower confidence needs a smaller $z^*$, so the interval is narrower.
To cut the margin of error in half, the sample size should be multiplied by about:
SE has $\displaystyle \sqrt{n}$ in the denominator, so $n$ must be 4 times as large.
A poll gives a $95\%$ confidence interval of $0.46$ to $0.54$ for the proportion supporting a measure. Can we conclude that a majority supports the measure?
The interval includes values below $0.50$, so a majority is not established.
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