z alpha/2 for 95 confidence interval
Understand the Problem
The question is asking for the critical value z for the two-tailed test associated with a 95% confidence interval. This value is used in statistics to determine the margin of error in a confidence interval calculation.
Answer
The critical value \( z \) is \( z = \pm 1.96 \).
Answer for screen readers
The critical value ( z ) for the two-tailed test associated with a 95% confidence interval is ( z = \pm 1.96 ).
Steps to Solve
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Identify the Confidence Level The given confidence level is 95%. This means we want to find the critical value that captures the central 95% of the standard normal distribution.
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Calculate the Tail Area Since this is a two-tailed test, we need to split the remaining area in the tails of the distribution. The total area under the standard curve is 1. Therefore, the area in each tail is: $$ \text{Tail area} = \frac{1 - 0.95}{2} = 0.025 $$
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Find the z-Score Now, we want to find the z-score that corresponds to an area of 0.025 in the left tail. We can use a z-table or standard normal distribution calculator to find this value.
Using the z-table, we look for the value that corresponds to 0.975 (which is $1 - 0.025$) since we need to cover the first 95% plus the lower tail area.
- Determine the Critical Values From the z-table, we find that the critical z-value for an area of 0.975 is approximately: $$ z = 1.96 $$ Therefore, for the two-tailed test, the critical values are $$ z = \pm 1.96 $$
The critical value ( z ) for the two-tailed test associated with a 95% confidence interval is ( z = \pm 1.96 ).
More Information
The critical value of ( z = 1.96 ) allows us to construct confidence intervals for population means when the population standard deviation is known or for large sample sizes. This value comes from the standard normal distribution and is widely used in hypothesis testing and confidence interval estimation.
Tips
- Confusing the z-value needed for one-tailed and two-tailed tests. Remember that a two-tailed test requires adding the tail areas together.
- Using the wrong area for critical value calculations. Make sure to subtract the confidence level from 1 and divide by 2 to find the tail area.
