Question

Find the critical value (z_e) necessary to form a confidence interval at the level of confidence shown below.

(C = 0.97)

(Round to two decimal places as needed.)

a) (1.88)

b) (1.96)

c) (2.05)

d) (2.17)

Answer 1

The critical value or z-score for a 97% confidence interval is approximately 2.17, which corresponds to the upper tail area of 0.015 in the standard normal distribution. This value is used in constructing the confidence interval.

Step-by-step explanation:

To find the critical value (ze) necessary to form a confidence interval with a confidence level of 0.97, we will refer to the z-score tables or use statistical software. The critical value is the z-score that corresponds to the desired confidence level, where there is an equal proportion of the standard normal distribution’s tail cut off at each end.

For a 97% confidence level, the tails on either side of the distribution would have an area of 1.5% each since (1 - 0.97) / 2 = 0.015. Looking up the corresponding z-score, we find that the critical value is approximately 2.17. Therefore, the correct critical value needed to construct a 97% confidence interval is 2.17, rounded to two decimal places.

To illustrate further, if you have a sample mean (x) and a known standard deviation (σ), and sample size (n), the confidence interval can be constructed using the formula:

x ± zeσ/√n