Question #335931

A sample of students from an introductory psychology class were polled regarding the number of hours they spent studying for the last exam. The data was used to make inferences regarding the other students taking the course. There data are below:



8 22.7 14.5 9 9 3.5 8 11 7.5 18



20 7.5 9 10.5 15 19 9 8.5 14 20



Compute a 99% confidence interval.




1
Expert's answer
2022-05-02T15:38:38-0400

n=20n=20


mean=xˉ=120(8+22.7+14.5+9+9mean=\bar{x}=\dfrac{1}{20}(8+ 22.7+ 14.5+ 9+9

+3.5+8+11+7.5+18+20+7.5+9+ 3.5+ 8+ 11+ 7.5+ 18+20+ 7.5 +9

+10.5+15+19+9+8.5+14+20)=12.185+ 10.5 +15+ 19+ 9+ 8.5+ 14 +20)=12.185

s2=1201((812.185)2+(22.712.185)2s^2=\dfrac{1}{20-1}((8-12.185)^2+(22.7-12.185)^2

+(14.512.185)2+(912.185)2+(14.5-12.185)^2+(9-12.185)^2

+(912.185)2+(3.512.185)2+(9-12.185)^2+(3.5-12.185)^2

+(812.185)2+(1112.185)2+(8-12.185)^2+(11-12.185)^2

+(7.512.185)2+(1812.185)2+(7.5-12.185)^2+(18-12.185)^2

+(2012.185)2+(7.512.185)2+(20-12.185)^2+(7.5-12.185)^2

+(912.185)2+(10.512.185)2+(9-12.185)^2+(10.5-12.185)^2

+(1512.185)2+(1912.185)2+(15-12.185)^2+(19-12.185)^2

+(912.185)2+(8.512.185)2+(9-12.185)^2+(8.5-12.185)^2

+(1412.185)2+(2012.185)2)+(14-12.185)^2+(20-12.185)^2)

=28.542395=28.542395


s=28.5423955.3425083s=\sqrt{28.542395}\approx5.3425083

The critical value for α=0.01\alpha = 0.01 and df=n1=19df = n-1 = 19 degrees of freedom is tc=z1α/2;n1=2.860935.t_c = z_{1-\alpha/2; n-1} =2.860935. 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}})

=(12.1852.860935×5.342508320,=(12.185-2.860935\times\dfrac{5.3425083}{\sqrt{20}},

12.185+2.860935×5.34250832012.185+2.860935\times\dfrac{5.3425083}{\sqrt{20}}

=(8.767,15.603)=(8.767,15.603)

Therefore, based on the data provided, the 99% confidence interval for the population mean is 8.767<μ<15.603,8.767 < \mu < 15.603, which indicates that we are 99% 

confident that the true population mean μ\mu is contained by the interval (8.767,15.603).(8.767, 15.603).


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