Question

7. : A Gallup poll asked Americans how many books they read all or part of the way last year. The results for a random sample of 2006 Americans were an average of 18.8 books and a standard deviation of 13.4 books. Construct and interpret a 95% confidence interval for the true average number of books read by Americans last year.

Answer 1

Here in this scenario we have to calculate the Confidence Interval for population mean based on the sample information. Here notice that the population standard deviations is unknown.

To calculate the Confidence Interval for population mean we have to use t distribution because the population standard deviations is unknown but according to the central limit theorem we can use z distribution when the sample size is large enough.

In this scenario we can also use z distribution to calculate the Confidence Interval for population mean we also get same result.

Further we computed the 95% Confidence Interval for population mean using both methods t distribution and using z distribution.

The provided sample mean is X = 18.8 and the sample standard deviation is s = 13.4. The size of the sample is n= 2006 and the required confidence level is 95%. < The number of degrees of freedom are df = 2006 – 1 = 2005, and the significance level is = 0.05. a = 0.05 and df = 2005 degrees of Based on the provided information, the critical t-value for a = freedom is tc = 1.961. The 95% confidence for the population mean u is computed using the following expression te x s te x s CI= X X + vn vñ < Therefore, based on the information provided, the 95 % confidence for the population mean u is 1.961 x 13.4 CI = ( 18.8 = (18. 1.961 x 13.4 ,18.8+ V2006 V2006 o (18.8 – 0.587,18.8 +0.587) = (18.213, 19.387) = which completes the calculation. The provided sample mean is X = 18.8 and the population standard deviation is o = 13.4. The size of the sample is n = 2006 and the required confidence level is 95%. ( Based on the provided information, the critical z-value for a = 0.05 is ze = 1.96. The 95% confidence for the population mean u is computed using the following expression Z c Xo CI= (x Zc xo vn à + n Therefore, based on the information provided, the 95% confidence for the population mean u is 1.96 x 13.4 CI (18.8 - 1.96 x 13.4 2006 18.8 + V2006 o = (18.8 – 0.586, 18.8 + 0.586) = (18.214, 19.386) which completes the calculation.

The z critical value is calculated using Standerd normal z-table or using Excel and t critical value is calculated using t table or using Excel.

We noticed that the Confidence Interval in both methods are approximately same. So as sample size is large enough we can use z distribution according to central limit theorem.

Interpretation :

We are 95% Confident that the population mean is occupied by given Interval is 18.214 and 19.386.

Thank you.