Find The Critical Value Za/2 That Corresponds To A 90% Confidence Level.
The correct answer and explanation is :
Answer:
The critical value ( z_{\alpha/2} ) that corresponds to a 90% confidence level is 1.645.
Detailed Explanation:
In statistics, when constructing confidence intervals for a population mean (especially when the population standard deviation is known or the sample size is large), we often use the standard normal distribution (z-distribution). The critical value ( z_{\alpha/2} ) is a key part of building these intervals. It marks the point(s) on the z-distribution that capture the middle percentage of the curve corresponding to the desired confidence level.
Step-by-Step Process:
- Understand Confidence Level:
A 90% confidence level means that we want the middle 90% of the normal distribution curve to fall between two z-scores, one negative and one positive (because the standard normal curve is symmetric). - Find ( \alpha ):
In statistics, ( \alpha ) (alpha) represents the total probability of making a Type I error (rejecting a true null hypothesis). It is the area in the two tails of the normal distribution outside the confidence interval.
[
\alpha = 1 – \text{Confidence Level} = 1 – 0.90 = 0.10
] - Find ( \alpha/2 ):
Since the normal curve is symmetric, the probability is split equally between the two tails.
[
\alpha/2 = 0.10/2 = 0.05
]
So, each tail will have 5% of the area. - Find ( z_{\alpha/2} ):
We now need to find the z-value such that 5% of the area is to the right of it. Equivalently, 95% of the area is to the left of it (because 100% – 5% = 95%). - Using the Z-table:
Using a standard normal distribution table (or a calculator), we find that the z-score corresponding to an area of 0.9500 to the left is approximately 1.645.
Important Concept:
- A 90% confidence interval captures the middle 90% of data, meaning that 5% of the data lies in each tail.
- ( z_{\alpha/2} ) is positive 1.645 for the right side and negative 1.645 for the left side.
- In confidence interval formulas, we typically use the positive critical value.
Final Summary:
Thus, for a 90% confidence level, the critical value ( z_{\alpha/2} ) is 1.645.
This value is used to calculate margins of error and create confidence intervals for population parameters when the data distribution is normal or approximately normal.