Answer to Question #223438 in Statistics and Probability for Ruben

Question #223438

(a)   A study of 35 gamers showed that their average score on a particular game was 90 and the population standard deviation is 5.

(i)                    Find the best point estimate of the population mean.


 

(ii)                 Find the 95% confidence interval of the mean score for all gamers.




(iii)               Find the 95% confidence interval of the mean score if a sample of 70 gamers is used instead of a sample of 35.


 

(iv)               From your answer in part (ii) and (iii), which interval is smaller?



1
Expert's answer
2021-08-09T11:12:27-0400

(i) The best point estimate for the population mean is the sample mean: x=90.x=90.


(ii) The critical value for α=0.05\alpha=0.05  is zc=z1α/2=1.96.z_c=z_{1-\alpha/2}=1.96.

The corresponding confidence interval is computed as shown below:


CI=(xzc×σn,x+zc×σn)CI=(x-z_c\times\dfrac{\sigma}{\sqrt{n}}, x+z_c\times\dfrac{\sigma}{\sqrt{n}})

=(901.96×535,90+1.96×535)=(90-1.96\times\dfrac{5}{\sqrt{35}}, 90+1.96\times\dfrac{5}{\sqrt{35}})

=(88.344,91.656)=(88.344, 91.656)

Therefore, based on the data provided, the 95% confidence interval for the population mean is 88.344<μ<91.656,88.344<\mu<91.656, which indicates that we are 95% confident that the true population mean μ\mu

is contained by the interval (88.344,91.656).(88.344, 91.656).


(iii) The critical value for α=0.05\alpha=0.05  is zc=z1α/2=1.96.z_c=z_{1-\alpha/2}=1.96.

The corresponding confidence interval is computed as shown below:


CI=(xzc×σn,x+zc×σn)CI=(x-z_c\times\dfrac{\sigma}{\sqrt{n}}, x+z_c\times\dfrac{\sigma}{\sqrt{n}})

=(901.96×570,90+1.96×570)=(90-1.96\times\dfrac{5}{\sqrt{70}}, 90+1.96\times\dfrac{5}{\sqrt{70}})

=(88.829,91.171)=(88.829, 91.171)

Therefore, based on the data provided, the 95% confidence interval for the population mean is 88.829<μ<91.171,88.829<\mu<91.171, which indicates that we are 95% confident that the true population mean μ\mu

is contained by the interval (88.829,91.171).(88.829, 91.171).


(iv) The 95% confidence interval for n=70n=70 is narrower than the 95% confidence interval for n=35.n=35.




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