Answer: For 96% confidence interval for the population mean of miles driven :
Lower bound = 10,841 miles
Upper bound= 14,949 miles
Step-by-step explanation:
Here, population standard deviation is unknown and sample size is small , So the formula is used to find the confidence interval for
is given by :-
![\overline{x}\pm t^*\dfrac{s}{\sqrt{n}}](https://tex.z-dn.net/?f=%5Coverline%7Bx%7D%5Cpm%20t%5E%2A%5Cdfrac%7Bs%7D%7B%5Csqrt%7Bn%7D%7D)
, where n = sample size , = sample mean , t*= two tailed critical value s= sample standard deviation, .
Given,
=12,895 miles , s=3,801 miles, n=15 , degree of freedom = 14 [∵df=n-1]
For 96% confidence level , ![\alpha=0.04](https://tex.z-dn.net/?f=%5Calpha%3D0.04)
By t-distribution table ,
t-value(two tailed) for
and df =14 is t*=2.2638
Now , the 96% confidence interval for the population mean of miles driven will be :
![12895\pm (2.2638)\dfrac{3801 }{\sqrt{15}}\\\\=12895\pm (2.0930)(981.413)\approx12895\pm 2054=(12895- 2054,\ 12895+2054)\\\\=(10,841,\ 14,949)](https://tex.z-dn.net/?f=12895%5Cpm%20%282.2638%29%5Cdfrac%7B3801%20%7D%7B%5Csqrt%7B15%7D%7D%5C%5C%5C%5C%3D12895%5Cpm%20%282.0930%29%28981.413%29%5Capprox12895%5Cpm%202054%3D%2812895-%202054%2C%5C%2012895%2B2054%29%5C%5C%5C%5C%3D%2810%2C841%2C%5C%2014%2C949%29)
Hence, For 96% confidence interval for the population mean of miles driven :
lower bound = 10,841 miles
upper bound= 14,949 miles