Final answer:
For a 90% confidence interval, α is 10% with 5% in each tail of the normal distribution. For a 95% confidence interval, α is 5% with 2.5% in each tail. The z-score critical value for a 90% interval is 1.645 standard deviations from the mean.
Step-by-step explanation:
When constructing a 90% confidence interval for the mean, α represents the total area of the distribution that lies outside the confidence interval. It is the area that is not covered by the interval and is split equally between the two tails of the normal distribution. For a 90% confidence interval, α would be 10%, since 90% is within the interval. Therefore, there is 5% in each tail (α/2). Similarly, for constructing a two-sided 95% confidence interval, 5% of the probability is excluded from the confidence interval and a total of α = 0.05 is evenly split between the two tails, so there is 2.5% in each tail.
When we talk about the percentage of confidence intervals that contain the population mean μ, if we have correctly constructed our confidence intervals, approximately 90% of them would indeed contain the true mean if we used a 90% confidence level. This established reliability is due to the statistical methods used to construct the intervals.
The critical value needed for a 90% confidence interval would be the z-score that captures the central 90% of the probability distribution, which is 1.645. This value marks the boundary on either end of the confidence interval, and going out 1.645 standard deviations from the sample mean captures the desired central 90% of the data.