A random sample of 90 observations produced a mean x̄ = 25.9 and a standard deviation s = 2.7. How do you find a 95% confidence interval for the population mean μ?

A 90% confidence interval for the population mean μ? A 99% confidence interval for the population mean μ?

1 Answer
Jan 29, 2017

The 95% confidence interval for #mu# is #(25.33, 26.47)#.
The 90% confidence interval is #(25.48, 26.37)#.
The 99% confidence interval is #(25.15, 26.65)#.

Explanation:

The formula for a #100(1-\alpha)%# confidence interval for #\mu# is

#bar x+-(t_(\alpha//2," "n"-1") xx s/sqrtn)#

where

  • #bar x# is our sample mean,
  • #t_(\alpha//2," "n"-1")# is the point on the #t#-distribution (with #n-1# degrees of freedom) with #100(\alpha/2)%# of the distribution's area to its right,
  • #s# is the sample standard deviation, and
  • #n# is the sample size.

Note: this formula assumes the population size #N# is unknown (or at least sufficiently large relative to #n#).

For a 95% confidence interval, #\alpha=0.05#, because

#100(1-0.05)%#
#=100(0.95)%#
#=95%#.

To compute the confidence interval desired, we simply plug in our values (and, in the case of the #t_(\alpha//2)# value, look it up) and simplify:

#color(white)=bar x+-(t_(\alpha//2," "n"-1") xx s/sqrtn)#
#=25.9+-(t_(0.025,89) xx 2.7/sqrt90)#
#=25.9+-(1.987 xx 0.2846)#
#=25.9+-(0.5655)#
#=(25.33, 26.47)#

While #s# itself represents the estimate for the population standard deviation, #s/sqrtn# represents the standard error of our estimate #bar x#—that is, it measures how far from #mu# our estimate #bar x# is likely to be. As our sample size #n# grows larger, our estimate #bar x# gets more precise, and so the standard error shrinks.

#t_(\alpha//2," "n"-1")# is like a scale factor which determines how many standard errors wide our margin of error will be. The more confident we wish to be about the interval including #mu#, the higher this #t#-value needs to be. That is, a smaller #\alpha# means a larger #t_(\alpha//2).#

To obtain different confidence intervals for #mu#, simply look up the different value necessary for #t_(\alpha//2," "n"-1")# and plug it into the formula, leaving all other values the same. I'll leave the calculation of the 90% C.I. and 99% C.I. as an exercise.