The Greater Pittsburgh Area Chamber of Commerce wants to estimate the mean time workers who are employed in the downtown area spend getting to work. A sample of 15 workers reveals the following number of minutes spent traveling.
i) Develop a 90%, 95% and 98 percent confidence interval for the population mean?
ii) Compare and interpret the result?
1
Expert's answer
2021-06-16T09:28:17-0400
29,38,38,33,38,21,45,34,40,37,37,42,30,29,35
n=15
Sample mean
xˉ=ni=1∑nxi=151(29+38+38+33+38+21
+45+34+40+37+37+42+30+29+35)
=15526≈35.066667
Var(x)=s2=n−1i=1∑n(xi−xˉ)2
=15−11((29−15526)2+(38−15526)2
+(38−15526)2+(33−15526)2+(38−15526)2
+(21−15526)2+(45−15526)2+(34−15526)2
+(40−15526)2+(37−15526)2+(37−15526)2
+(42−15526)2+(38−15526)2+(29−15526)2
+(35−15526)2≈36.2095238
s=s2≈6.017435
i)
90%
The critical value for α=0.1 and df=n−1=15−1=14 degrees of freedom is tc=1.76131. The corresponding confidence interval is computed as shown below:
CI=(xˉ−tc×ns,xˉ+tc×ns)
=(35.066667−1.76131×156.017435,
35.066667+1.76131×156.017435)
≈(32.33013,37.80321)
Therefore, based on the data provided, the 90% confidence interval for the population mean is 32.33013<μ<37.80321, which indicates that we are 90% confident that the true population mean μ is contained by the interval (32.33013,37.80321).
95%
The critical value for α=0.05 and df=n−1=15−1=14 degrees of freedom is tc=2.144787. The corresponding confidence interval is computed as shown below:
CI=(xˉ−tc×ns,xˉ+tc×ns)
=(35.066667−2.144787×156.017435,
35.066667+2.144787×156.017435)
≈(31.73432,38.39901)
Therefore, based on the data provided, the 95% confidence interval for the population mean is 31.73432<μ<38.39901, which indicates that we are 95% confident that the true population mean μ is contained by the interval (31.73432,38.39901).
98%
The critical value for α=0.02 and df=n−1=15−1=14 degrees of freedom is tc=2.624494. The corresponding confidence interval is computed as shown below:
CI=(xˉ−tc×ns,xˉ+tc×ns)
=(35.066667−2.624494×156.017435,
35.066667+2.624494×156.017435)
≈(30.98900,39.14433)
Therefore, based on the data provided, the 98% confidence interval for the population mean is 30.98900<μ<39.14433, which indicates that we are 98% confident that the true population mean μ is contained by the interval (30.98900,39.14433).
ii)
The 90% confidence interval is (32.33013,37.80321).
The 95% confidence interval is (31.73432,38.39901).
The 98% confidence interval is (30.98900,39.14433).
The 95% confidence interval is wider than the 90% confidence interval.
The 98% confidence interval is wider than the 95% confidence interval and wider than the 90% confidence interval.
Increasing the confidence level increases the error bound, making the confidence interval wider.
Decreasing the confidence level decreases the error bound, making the confidence interval narrower.
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