Question

Which of the following is true about a 95% confidence interval, with a known sigma?

a. The corresponding value for z is 1.65

b. 95% of similarly constructed intervals would contain the population mean.

c. we can use either the t or z distribution, whichever we prefer

d. All options are true

Answer 1

Solution

Back-up Theory

100(1 - α) % Confidence Interval for μ, when σ known is: Xbar ± (Z_{α /2})σ/√n …….(1)

Where Xbar = sample mean, Z_{α /2} = upper (α /2)% point of N(0, 1), σ = population standard deviation and n = sample size.

If (a, b) is a 100(1 - α) % Confidence Interval for a (population) parameter, say μ, then 95% of similarly constructed intervals would contain the population parameter μ . ……. (2)

Now, to work out the answer,

Vide (1), the corresponding value for z must be upper 2.5% point of N(0, 1), which is 1.96 and NOT 1.645 [1.65 is the upper 5% point of N(0, 1)].

Since σ is known, the distribution is Normal (i.e., z distribution) [If σ is unknown, the distribution is t.]

Vide (2), option b is correct.

Thus, the correct answer is Option b. ANSWER