# 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?

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

The 95% confidence interval for

The 90% confidence interval is

The 99% confidence interval is

#### Explanation:

The formula for a

#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,

#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

#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 **standard error** of our estimate

To obtain different confidence intervals for