Answer:
c) there is 95% confidence that the population mean number of books read is between 13.77 and 15.83.
Step-by-step explanation:
We have to calculate a 95% confidence interval for the mean.
The population standard deviation is not known, so we have to estimate it from the sample standard deviation and use a t-students distribution to calculate the critical value.
The sample mean is M=14.8.
The sample size is N=1003.
When σ is not known, s divided by the square root of N is used as an estimate of σM:
![s_M=\dfrac{s}{\sqrt{N}}=\dfrac{16.6}{\sqrt{1003}}=\dfrac{16.6}{31.67}=0.524](https://tex.z-dn.net/?f=s_M%3D%5Cdfrac%7Bs%7D%7B%5Csqrt%7BN%7D%7D%3D%5Cdfrac%7B16.6%7D%7B%5Csqrt%7B1003%7D%7D%3D%5Cdfrac%7B16.6%7D%7B31.67%7D%3D0.524)
The degrees of freedom for this sample size are:
The t-value for a 95% confidence interval and 1002 degrees of freedom is t=1.96.
The margin of error (MOE) can be calculated as:
Then, the lower and upper bounds of the confidence interval are:
![LL=M-t \cdot s_M = 14.8-1.03=13.77\\\\UL=M+t \cdot s_M = 14.8+1.03=15.83](https://tex.z-dn.net/?f=LL%3DM-t%20%5Ccdot%20s_M%20%3D%2014.8-1.03%3D13.77%5C%5C%5C%5CUL%3DM%2Bt%20%5Ccdot%20s_M%20%3D%2014.8%2B1.03%3D15.83)
The 95% confidence interval for the mean number of books read is (13.77, 15.83).
This indicates that there is 95% confidence that the true mean is within 13.77 and 15.83. Also, that if we take multiples samples, it is expected that 95% of the sample means will fall within this interval.