Question

People were polled on how many books they read the previous year. Initial survey results indicate that sequals16.3 books. Complete parts ​(a) through ​(d) below. LOADING... Click the icon to view a partial table of critical values. ​(a) How many subjects are needed to estimate the mean number of books read the previous year within four books with 90​% ​confidence? This 90 % confidence level requires nothing subjects. ​(Round up to the nearest​ subject.) ​(b) How many subjects are needed to estimate the mean number of books read the previous year within two books with 90​% ​confidence? This 90 % confidence level requires nothing subjects. ​(Round up to the nearest​ subject.) ​(c) What effect does doubling the required accuracy have on the sample​ size? A. Doubling the required accuracy nearly doubles the sample size. B. Doubling the required accuracy nearly halves the sample size. C. Doubling the required accuracy nearly quarters the sample size. D. Doubling the required accuracy nearly quadruples the sample size. ​(d) How many subjects are needed to estimate the mean number of books read the previous year within four books with 99​% ​confidence? This 99​% confidence level requires nothing subjects. ​(Round up to the nearest​ subject.) Compare this result to part ​(a). How does increasing the level of confidence in the estimate affect sample​ size? Why is this​ reasonable? A. Increasing the level of confidence increases the sample size required. For a fixed margin of​ error, greater confidence can be achieved with a smaller sample size. B. Increasing the level of confidence decreases the sample size required. For a fixed margin of​ error, greater confidence can be achieved with a larger sample size. C. Increasing the level of confidence increases the sample size required. For a fixed margin of​ error, greater confidence can be achieved with a larger sample size. D. Increasing the level of confidence decreases the sample size required. For a fixed margin of​ error, greater confidence can be achieved with a smaller sample size.

Answer 1

a)

Standard Deviation ,   σ =    16.3                  
sampling error ,    E =   4                  
Confidence Level ,   CL=   90%                  
                          
alpha =   1-CL =   10%                  
Z value =    Zα/2 =    1.645   [excel formula =normsinv(α/2)]              
                          
Sample Size,n = (Z * σ / E )² = (   1.645   *   16.3   /   4   ) ² =   44.927
                          
                          
So,Sample Size needed=       45                  

b)

Standard Deviation ,   σ =    16.3                  
sampling error ,    E =   2                  
Confidence Level ,   CL=   90%                  
                          
alpha =   1-CL =   10%                  
Z value =    Zα/2 =    1.645   [excel formula =normsinv(α/2)]              
                          
Sample Size,n = (Z * σ / E )² = (   1.645   *   16.3   /   2   ) ² =   179.709
                          
                          
So,Sample Size needed=       180                  

c)  Doubling the required accuracy nearly quadruples the sample size.

d) Standard Deviation ,   σ =    16.3                  
sampling error ,    E =   4                  
Confidence Level ,   CL=   99%                  
                          
alpha =   1-CL =   1%                  
Z value =    Zα/2 =    2.576   [excel formula =normsinv(α/2)]              
                          
Sample Size,n = (Z * σ / E )² = (   2.576   *   16.3   /   4   ) ² =   110.177
                          
                          
So,Sample Size needed=       111                  

Increasing the level of confidence increases the sample size required. For a fixed margin of​ error, greater confidence can be achieved with a larger sample size.

e)