Interval endpoints, sample size and interpretation.
Example 1
A random sample of 64 from a normal population has mean 53; population standard deviation is known to be 24. Using z∗=1.96, find the LOWER endpoint of a 95% confidence interval for the population mean. Give your answer to 2 decimal places.
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Answer: 47.12
- SE=√6424=3.
- Margin =1.96×3.
- Lower endpoint =53−5.88.
- Rounded answer: 47.12.
Example 2
Population standard deviation is known to be 10. With z∗=1.96, what minimum integer sample size gives margin of error at most 6?
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Answer: 11
- Require √n1.96σ≤6.
- n≥(1.96×610)2=10.6711111111.
- Round UP to 11.
Example 3
A 95% confidence procedure gives the interval (52,58) for a population mean. Which interpretation of 95% is correct?
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Answer: About 95% of repeated intervals contain the population mean
- The confidence level describes the long-run coverage of the method.
- The population mean is fixed; the interval varies with the sample.