Find the critical value Za/2 that corresponds to the given confidence level. 90% … Za12 = (Round to two decimal places as needed.)
The Correct Answer and Explanation is:
To find the critical value ( Z_{\alpha/2} ) for a 90% confidence level, we first need to understand what this critical value represents. The critical value is the z-score that corresponds to the tails of the standard normal distribution in the context of constructing confidence intervals.
Steps to Determine ( Z_{\alpha/2} )
- Determine the significance level (( \alpha )):
- The confidence level of 90% implies that 10% of the distribution is not covered by the confidence interval. This 10% is the significance level ( \alpha ):
[
\alpha = 1 – \text{Confidence Level} = 1 – 0.90 = 0.10
]
- Calculate ( \alpha/2 ):
- Since the confidence interval splits the significance level between the two tails of the normal distribution, we need to find ( \alpha/2 ):
[
\frac{\alpha}{2} = \frac{0.10}{2} = 0.05
]
- Find the critical value ( Z_{\alpha/2} ):
- The next step is to find the z-score that corresponds to the cumulative area of ( 1 – \frac{\alpha}{2} ) (or 0.95). This value represents the point below which 95% of the data lies in the standard normal distribution:
[
1 – 0.05 = 0.95
]
- Using the Z-table or a calculator:
- We can use the standard normal (Z) distribution table or a statistical calculator to find the z-score corresponding to a cumulative probability of 0.95. Looking at a Z-table or using a calculator, we find:
[
Z_{0.05} \approx 1.645
]
- Round the answer:
- The critical value ( Z_{\alpha/2} ) is therefore:
[
Z_{\alpha/2} = 1.65 \text{ (rounded to two decimal places)}
]
Conclusion
Thus, the critical value ( Z_{\alpha/2} ) for a 90% confidence level is approximately 1.65. This value is essential for constructing confidence intervals, as it helps to determine the margin of error for estimates based on sample data. It indicates how far from the sample mean we can expect to extend our interval to capture the true population parameter with the specified level of confidence. Understanding these concepts is crucial in statistical inference, where making informed conclusions about a population based on sample data is fundamental.