Question

A 90% confidence interval for a population mean, μ, is given as (21.73,
24.64). This confidence interval is based on a simple random sample of 41
observations. Calculate the sample mean and standard deviation. Assume
that all conditions necessary for inference are satisfied. Use the t-
distribution in any calculations.

Answer 1

90% confidence Interval for mean is given by:

( , )

where alpha = (1- confidence Interval)/100

   n is number of observations

s is standard deviation

  talpha/2 , n-1 is t value at n-1 degree of freedom

But this Interval is given to us as :

(21.73 , 24.64)

Thus ,

= 21.73

= 24.64

Now , for 90% confidence Interval

alpha = 10% or 0.10

n-1 = 41 - 1 = 40

Now using Appendix table A.5 ,

talpha/2 , n-1 = t0.10/2 , 40

= 1.684

Thus , = 21.73

= 24.64

Adding both above equations and get:

+    = 21.73 + 24.64

2 mean = 46.37

Thus, mean = 23.185

Substitute mean value to any one of the above equation:

   = 24.64

23.185 + 1.684(s/) = 24.64

1.684(s/) = 1.455

s = (1.455/1.684)*

s = (0.864)*

s = 5.5324

Thus , standard deviation = 5.5324

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