Question

For a normal population with the known variance σ2, answer the following questions. Round your answers to two decimal places (e.q

a) What value of confidence level gives Za/2 equal to 2.05?
b) What value of confidence level gives Za/2 equal to 2.37?
c) what value of confidence level gives Zn2 equal to 1.13?

Answer 1

The confidence level can be determined from the given Z-scores by finding the cumulative area to the left of the Z-score and calculating 1 - (2 * the area in the tail). For Za/2 values of 2.05, 2.37, and 1.13, the confidence levels are approximately 95.96%, 98.22%, and 74.16%, respectively.

Step-by-step explanation:

The confidence level (CL) corresponds to the central area under the standard normal distribution curve. For a given value of Za/2, we can find the associated confidence level by considering the area to the left of Za/2, which is 1 - (a/2), and then computing 1 - a to get the confidence level.

a) To find the confidence level for Za/2 = 2.05, we can look up the area to the left of Z = 2.05 in a standard normal distribution table or use a calculator to get approximately 0.9798, which represents 97.98% of the area under the curve. Thus, a/2 = 0.0202 and CL = 1 - (2 * 0.0202) = 0.9596 or 95.96%.

b) For Za/2 = 2.37, the area to the left of Z = 2.37 is roughly 0.9911, corresponding to 99.11% of the area under the curve. Hence, a/2 = 0.0089 and CL = 1 - (2 * 0.0089) = 0.9822 or 98.22%.

c) The third part of the question contains a typo: 'Zn2'. Assuming the intended term is Za/2 and is equal to 1.13, the area to the left is approximately 0.8708. Therefore, a/2 = 0.1292 and CL = 1 - (2 * 0.1292) = 0.7416 or 74.16%.

Answer 2

a) The confidence level that gives Zₐ/₂ equal to 2.05 is approximately 95.53%.

b) The confidence level that gives Zₐ/₂ equal to 2.37 is approximately 97.72%.

c) The confidence level that gives Zₙ/₂ equal to 1.13 is approximately 87.81%.

Step-by-step explanation:

In statistics, the Z-score is a measure of how many standard deviations a particular data point is from the mean of a normal distribution. The formula for calculating the confidence level (CL) given a Z-score (Z) is CL = 1 - α, where α is the significance level. The Zₐ/₂ value corresponds to the critical Z-score for a two-tailed test with a significance level of α/2.

a) For Zₐ/₂ equal to 2.05, we find the corresponding confidence level using the formula CL = 1 - α. Solving for α/2 gives us α/2 = 0.025. Therefore, the confidence level is 1 - 0.025 - 0.975 or approximately 97.5%, and rounding to two decimal places, the answer is 95.53%.

b) Similarly, for Zₐ/₂ equal to 2.37, we find α/2 = 0.0185. The confidence level is then 1 - 0.0185 - 0.9815 or approximately 98.15%, rounded to two decimal places, resulting in a confidence level of 97.72%.

c) In the case of Zₙ/₂ equal to 1.13, we use the same formula to find α/2. Solving for α/2, we get α/2 = 0.1292. The confidence level is then 1 - 0.1292 - 0.8708 or approximately 87.08%, rounded to two decimal places, resulting in a confidence level of 87.81%.