Construct intervals, estimate sample size and assess claims.
Example 1
A random sample of 400 people contains 160 successes. Using z∗=1.96 and the normal approximation, find the UPPER endpoint of a 95% confidence interval for the population proportion. Give your answer to 4 decimal places.
Show worked answer
Answer: 0.4480
- p^=400160=0.4.
- SE estimate =√0.44001−0.4.
- Upper endpoint =p^+1.96×SE.
- Rounded answer: 0.4480.
Example 2
With no prior proportion estimate, use p=0.5 and z∗=1.96. Find the minimum integer sample size for a margin of error at most 0.02.
Show worked answer
Answer: 2401
- Use n≥E2z∗2p(1−p).
- n≥1.962×0.0220.25=2401.
- Round UP to 2401.
Example 3
A 95% confidence interval for a population proportion is [0.2481,0.3519]. A proposed value is 0.4019. Is that value compatible with this interval?
Show worked answer
Answer: No: it is above the interval
- Compare 0.4019 with both endpoints, 0.2481 and 0.3519.
- No: it is above the interval. An interval comparison gives evidence, not certainty about the claim.