find z a/2 for a 99% confidence interval.
The Correct Answer and Explanation is:
To find the value of zα/2z_{\alpha/2}zα/2 for a 99% confidence interval, we are looking for the critical value of the standard normal distribution that corresponds to the upper tail probability of α/2\alpha/2α/2, where α\alphaα is the significance level.
Steps:
- Determine the confidence level and significance level:
- The given confidence level is 99%, meaning the area under the normal curve that corresponds to the middle of the distribution is 0.99.
- The significance level α\alphaα is the complement of the confidence level: α=1−0.99=0.01\alpha = 1 – 0.99 = 0.01α=1−0.99=0.01
- Since the confidence interval is two-tailed, we divide the significance level by 2: α/2=0.01/2=0.005\alpha/2 = 0.01/2 = 0.005α/2=0.01/2=0.005 This means that there is 0.5% of the distribution in the upper tail and 0.5% in the lower tail.
- Look up the value of zα/2z_{\alpha/2}zα/2:
- The zα/2z_{\alpha/2}zα/2 value corresponds to the z-score that leaves 0.005 in the upper tail of the standard normal distribution. This is equivalent to finding the z-score where the cumulative probability is 1−0.005=0.9951 – 0.005 = 0.9951−0.005=0.995.
- From standard normal distribution tables, or using a calculator, the z-score that corresponds to 0.995 cumulative probability is approximately 2.576.
Conclusion:
For a 99% confidence interval, the value of zα/2z_{\alpha/2}zα/2 is 2.576.
This value is used in calculating the margin of error for a confidence interval, along with the standard deviation (or standard error) of the sample, to determine the range within which the population parameter is likely to fall. In practice, the formula for a confidence interval would be:Confidence Interval=μ^±zα/2×σn\text{Confidence Interval} = \hat{\mu} \pm z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}Confidence Interval=μ^±zα/2×nσ
where:
- μ^\hat{\mu}μ^ is the sample mean,
- σ\sigmaσ is the population standard deviation (or sample standard deviation),
- nnn is the sample size.
