Help!!! Confidence intervals

Question

A sample of size n=78  is drawn from a normal population whose standard deviation is =σ6.2 The sample mean is x= 39.06. Construct an 80% confidence interval for μ.

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Answer 1

Since we know the sample is pulled from a normal population with a known standard deviation, we can assume the sample is normally distributed. This allows us to use a z-score instead of a t-score for the critical value. This critical value comes out to be approximately 1.28 from the table (1.281552 when using a calculator). Then the margin of error (MOE) equals that multiplied by standard deviation of the mean:

MOE = z_α/2•σ/√n

= 1.28•6.2/√78

= 0.8986

Lastly the confidence interval is found by subtracting and adding this from the mean.

CI = (µ-MOE, µ+MOE)

= (39.06-0.8986, 39.06+0.8986)

= (38.1614, 39.9586)

Thus, 38.16 < µ < 39.96.