Question #336505

A random sample is drawn from a population of known standard deviation 22.1. Construct a 95% confidence interval for the population mean based on the information given (not all of the information given need be used).

a. n=121, x¯=82.4, s=21.9 n=121, x¯=82.4, s=21.9

b. n=81, x¯=82.4, s=21.9 n=81, x¯=82.4, s=21.9


1
Expert's answer
2022-05-03T12:40:14-0400

The corresponding confidence interval is computed as shown below:


CI=(xˉtc×sn,xˉ+tc×sn)CI=(\bar{x}-t_c\times\dfrac{s}{\sqrt{n}}, \bar{x}+t_c\times\dfrac{s}{\sqrt{n}})

a) The critical value for α=0.05,df=n1=120\alpha = 0.05, df=n-1=120 degrees of freedom is tc=z1α/2;n1=1.97993.t_c=z_{1−α/2;n−1}=1.97993.


CI=(82.41.97993×21.9121,CI=(82.4-1.97993\times\dfrac{21.9}{\sqrt{121}},

82.4+1.97993×21.9121)82.4+1.97993\times\dfrac{21.9}{\sqrt{121}})

=(78.458,86.342)=(78.458,86.342)

Therefore, based on the data provided, the 95% confidence interval for the population mean is 78.458<μ<86.342,78.458< \mu < 86.342, which indicates that we are 95% confident that the true population mean μ\mu is contained by the interval (78.458,86.342).(78.458,86.342).


b) The critical value for α=0.05,df=n1=80\alpha = 0.05, df=n-1=80 degrees of freedom is tc=z1α/2;n1=1.990063.t_c=z_{1−α/2;n−1}=1.990063.

CI=(82.41.990063×21.981,CI=(82.4-1.990063\times\dfrac{21.9}{\sqrt{81}},

82.4+1.990063×21.981)82.4+1.990063\times\dfrac{21.9}{\sqrt{81}})

=(77.558,87.242)=(77.558,87.242)

Therefore, based on the data provided, the 95% confidence interval for the population mean is 77.558<μ<87.242,77.558< \mu < 87.242, which indicates that we are 95% confident that the true population mean μ\mu is contained by the interval (77.558,87.242).(77.558,87.242).



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