Final answer:
To construct a confidence interval, find the point estimate, determine the distribution, find the critical z-score for the confidence level, calculate the error bound EBM, and then construct the interval by adding and subtracting the margin of error from the point estimate. A 95% confidence interval uses approximately 1.96 as the z-score.
Step-by-step explanation:
Constructing a Confidence Interval
To construct a confidence interval for a population parameter, the following steps are generally followed:
- Find the point estimate of the parameter.
- Determine the appropriate distribution to use, which is often the normal distribution if the sample size is large, or the t-distribution if the sample size is small and the population standard deviation is unknown.
- Find the critical z-score or t-score that corresponds to the desired confidence level. For a 90% confidence level, you would find the z-score that leaves 5% in each tail of the normal distribution.
- Calculate the error bound EBM, which is the margin of error for the estimate.
- Construct the confidence interval by adding and subtracting the error bound from the point estimate.
The confidence interval has the format of the sample mean (μ) plus or minus the margin of error (ME). When constructing a 95% confidence interval, you would use a z-score that corresponds to 2.5% in each tail, which is approximately 1.96. The final interval is μ ± ME.
For the example of determining the population proportion of people who believe the president is doing an acceptable job:
- State the confidence interval: The calculated interval using the sample proportion and the error bound.
- Sketch the graph: A normal distribution with the point estimate and the bounds marked.
- Calculate the error bound: The product of the critical z-score and the standard error of the proportion.