Question

Consider a normal population distribution with the value of known. A button hyperlink to the SALT program that reads: Use SALT.

(a) What is the confidence level for the interval x ± 2.88⁄ n ? (Round your answer to one decimal place.) %
(b) What is the confidence level for the interval x ± 1.41⁄ n ? (Round your answer to one decimal place.) %
(c) What value of z⁄2 in the CI formula below results in a confidence level of 99.7%? (Round your answer to two decimal places.) x − z⁄2 · n , x + z⁄2 · n z⁄2 =
(d) Answer the question posed in part (c) for a confidence level of 78%. (Round your answer to two decimal places.)

Answer 1

The confidence interval at a critical value of 2.88 corresponds to a 99.6% confidence level, and at 1.41, it's approximately 84.1%. For a 99.7% confidence level, the z-score is about 3.00, and for a 78% confidence level, it's approximately 1.23.

Step-by-step explanation:

The answers to the questions regarding confidence intervals for a normal population distribution are:

• (a) To find the confidence level for the interval x ± 2.88√÷ n, we need to determine the z-score that corresponds to this critical value. Using a standard normal distribution table or calculator, we find that a z-score of 2.88 corresponds to a confidence level of approximately 99.6%. Therefore, the confidence level is 99.6% (rounded to one decimal place).
• (b) Similarly, a z-score of 1.41 corresponds to a confidence level of approximately 84.1%. So, the confidence level is 84.1% (rounded to one decimal place).
• (c) For a confidence level of 99.7%, we are looking for the z-score that corresponds to the middle 99.7% of the distribution, leaving 0.3% in the tails. The two tails would each contain 0.15%, and the corresponding z-score is approximately 3.00. Thus, z√÷ = 3.00 (rounded to two decimal places).
• (d) For a confidence level of 78%, we are looking for a z-score that corresponds to the middle 78% of the distribution, leaving 22% in the tails. The two tails would each contain 11%, and the corresponding z-score is approximately 1.23. Thus, z√÷ = 1.23 (rounded to two decimal places).