Question

People were polled on how many books they read the previous year. Initial survey results indicate that sequals16.2 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 six 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 three 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 quadruples the sample size. D. Doubling the required accuracy nearly quarters the sample size. ​(d) How many subjects are needed to estimate the mean number of books read the previous year within six 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 decreases the sample size required. For a fixed margin of​ error, greater confidence can be achieved with a larger 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 smaller 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 increases 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 =    16.2
sampling error ,    E =   6
Confidence Level ,   CL=   90%
      
alpha =   1-CL =   10%
Z value =    Zα/2 =    1.6449 [excel formula =normsinv(α/2) ]
      
Sample Size,n =    (Z*σ / E)² =   19.7234
      
      
So,Sample Size needed=       20

b)

Standard Deviation ,   σ =    16.2  
sampling error ,    E =   3  
Confidence Level ,   CL=   90%  
          
alpha =   1-CL =   10%  
Z value =    Zα/2 =    1.6449   [excel formula =normsinv(α/2) ]
          
Sample Size,n =    (Z*σ / E)² =   78.8936  
          
          
So,Sample Size needed=       79  

c)

Doubling the required accuracy nearly quadruples the sample size.

As we have to estimate the mean with high accuracy i.e. within 3 books the sample size is 79. When we had to estimate the mean with a slightly lower accuracy i.e. within 6 books the sample size was only 20. i.e. as the accuracy doubles the sample size becomes 4 times

d)

Standard Deviation ,   σ =    16.2  
sampling error ,    E =   6  
Confidence Level ,   CL=   99%  
          
alpha =   1-CL =   1%  
Z value =    Zα/2 =    2.5758   [excel formula =normsinv(α/2) ]
          
Sample Size,n =    (Z*σ / E)² =   48.3684  
          
          
So,Sample Size needed=       49  
e)

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.

This is simply because to achieve a greater confidence we need to take a larger sample.