For a confidence level of 85%, find the critical value
The correct answer and explanation is:
To find the critical value for a confidence level of 85%, we are typically referring to a z-score (when population standard deviation is known or large sample size) or t-score (when standard deviation is unknown and sample size is small). For this explanation, we will assume the standard normal distribution (z-distribution).
✅ Correct Answer:
The critical value (z*) for an 85% confidence level is approximately:
z* = ±1.44
📘 Explanation (300 words):
In statistics, a confidence level is the percentage of all possible samples that can be expected to include the true population parameter. An 85% confidence level means that we are 85% confident that the true population parameter lies within the confidence interval.
To calculate the critical value for a confidence level using the standard normal (z) distribution, follow these steps:
Step 1: Understand the Confidence Level
An 85% confidence level means that the middle 85% of the normal distribution is between two z-scores, with 15% left in the tails (7.5% in each tail because it’s a two-tailed test).
Step 2: Find the Area in One Tail
- Total area in both tails = 100% − 85% = 15%
- Area in one tail = 15% ÷ 2 = 7.5% = 0.075
So, we are looking for the z-value where the area to the left is:
1 − 0.075 = 0.925
Step 3: Use the Standard Normal Table or Calculator
Look up the z-score that corresponds to 0.925 cumulative area. This value is:
z ≈ 1.44
Step 4: Interpret the Result
Since it’s a two-tailed test, the critical values are:
±1.44
This means if you were constructing a confidence interval, you would go 1.44 standard deviations above and below the sample mean.
🔑 Summary:
- Confidence level: 85%
- Tails: 7.5% in each
- Cumulative area to left: 0.925
- Critical value (z*) = ±1.44
This critical value helps construct the confidence interval or perform hypothesis tests with 85% confidence.