Which z-score is used in a (a) 90%, (b) 98%, and (c) 99.9% confidence interval for a population proportion? a. The Z-score used in a 90% confidence interval for a population proportion is 1 (Round to two decimal places as needed.) b. The Z-score used in a 98% confidence interval for a population proportion is (Round to two decimal places as needed.) c. The Z-score used in a 99.9% confidence interval for a population proportion is (Round to two decimal places as needed.)
The Correct Answer and Explanation is :
The critical z-scores for different confidence intervals are as follows:
a. 90% confidence interval → 1.645
b. 98% confidence interval → 2.33
c. 99.9% confidence interval → 3.29
Explanation:
The z-score (also known as the critical value) represents the number of standard deviations a value is from the mean in a standard normal distribution. In confidence interval estimation, it is used to determine the margin of error around a sample statistic when estimating a population parameter.
Understanding How Z-Scores Are Found
To determine the z-score for a confidence interval, we look at the area under the normal curve corresponding to the desired confidence level. This area represents the proportion of values that fall within the given range.
- 90% Confidence Interval
- A 90% confidence interval means 90% of the data falls within this range, leaving 5% in each tail (since the normal distribution is symmetrical).
- We look up the cumulative probability of 0.9500 (since 90% is in the middle, the remaining 10% is split equally, leaving 5% in the upper tail).
- This corresponds to z = 1.645.
- 98% Confidence Interval
- Here, 98% of the data is within the interval, leaving 1% in each tail.
- Looking up the cumulative probability of 0.9900, the z-score is 2.33.
- 99.9% Confidence Interval
- This confidence level leaves 0.05% in each tail.
- The cumulative probability for 99.95% is 3.29.
Conclusion
The higher the confidence level, the wider the confidence interval and the larger the z-score. This means that as we increase confidence, we need to account for more variability, making the interval wider.