Question

Looking at my answers, I was am always off buy about range of .1 to .4 from the correct answer that is stated. I am having a really tough time trying to figure out how to get my number to be exactly like what is shown as correct so I need a little help with figuring out what I am doing wrong. Here is an example of a question I just cant seem to figure out. I'll show all my steps and let me know what I am doing wrong please

Construct a  90% confidence interval to estimate the population mean using the data below.
x̅ =50 σ=10 n=20 N=200

The  90% confidence interval for the population mean is (___, ___)
(Round to two decimal places as needed.)

My steps:
UCL_x̅ = x̅ + (critical z-value)(σ_x̅)
LCL_x̅ = x̅ - (critical z-value)(σ_x̅)

So I find σx̅

σ_x̅ = σ/√n σ_x̅ = 10/√20 = 2.236

Finding Critical z - value
α = (1 - confidence interval) then divide by 2 (1 - .90)/2 = .05 then using technology I find the z-value of .05 = 1.645

Plug it all in
50 + (1.645)(2.236) = 53.68
50 - (1.645)(2.236) = 46.32

I then have an answer that the 90% confidence interval for the population mean is (46.32, 53.68) which is incorrect. The correct answer shown is (46.50, 53.50) but I am unsure how to get this interval. Could someone work this out and show me the steps on how to get the correct interval please?

Answer 1

Note: The 90% confidence interval is (46.3217 , 53.6783) . Your answer and all the steps are correct. There may be some error in the given data or the answer.

We need to construct the 90% confidence interval for the population mean μ. The following information is provided:

x̅ =50, σ=10 and   n=20

The critical value for α=0.1 is Z_c​=Z_1−α/2 ​= 1.645. The corresponding confidence interval is computed as shown below: