Find the critical value za/2 that corresponds to the given confidence level.
80%
Za/2
The correct answer and explanation is :
Let’s solve this carefully:
The confidence level given is 80%.
The critical value ( z_{\alpha/2} ) corresponds to the value on the standard normal distribution such that the area between (-z_{\alpha/2}) and (z_{\alpha/2}) is the confidence level. In other words, you have 80% of the distribution’s area between (-z_{\alpha/2}) and (z_{\alpha/2}), leaving 20% of the area in the two tails combined (because 100% – 80% = 20%).
Since the two tails share the 20% equally, each tail has an area of:
[
\frac{20\%}{2} = 10\% = 0.10
]
So, the area to the left of ( z_{\alpha/2} ) must be:
[
1 – 0.10 = 0.90
]
Thus, we need to find the z-score that corresponds to a cumulative area of 0.90.
Using a standard normal distribution table or a calculator, the z-score for an area of 0.90 is approximately:
[
z_{\alpha/2} = 1.28
]
✅ Final Answer:
[
z_{\alpha/2} = 1.28
]
Detailed Explanation (300+ Words):
The concept of a critical value ( z_{\alpha/2} ) is fundamental when working with confidence intervals, particularly in statistics and inferential analysis. The critical value represents the number of standard deviations a data point is from the mean in a normal distribution. For a given confidence level, it marks the point beyond which a certain percentage of the distribution lies in the tails.
In this question, the confidence level is 80%, meaning we want the middle 80% of the normal distribution to be captured between (-z_{\alpha/2}) and (z_{\alpha/2}). To find ( z_{\alpha/2} ), we must first determine the total area left in the two tails of the distribution, which is 100% – 80% = 20%. Since the normal distribution is symmetric, this 20% is divided equally between the left and right tails, meaning each tail has 10% of the area.
To find the correct critical value, we look for the z-score that corresponds to an area of 90% (0.90) to the left of it. In a z-table (standard normal table), 0.90 corresponds approximately to a z-score of 1.28. This means 90% of the distribution lies to the left of 1.28, leaving 10% to the right, which matches the desired division of the tails.
Understanding critical values like ( z_{\alpha/2} ) is crucial because they directly influence the width of confidence intervals. A lower confidence level (like 80%) results in a smaller critical value and a narrower confidence interval, while a higher confidence level (like 95% or 99%) leads to a larger critical value and a wider interval. This trade-off is important in research, where precision and certainty must be balanced.
Thus, for an 80% confidence level, the corresponding critical value ( z_{\alpha/2} ) is 1.28.