People were polled on how many books they read the previous year. Initial survey

Question

People were polled on how many books they read the previous year. Initial survey results indicate that sequals 12.4 books. Complete parts (a) through (d) below. (a) How many subjects are needed to estimate the mean number of books read the previous year within six books with 95 % confidence? This 95 % 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 95 % confidence? This 95 % 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 quadruples the sample size. B. Doubling the required accuracy quarters the sample size. C. Doubling the required accuracy doubles the sample size. D. Doubling the required accuracy halves 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 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.

Explanation / Answer

a)

Note that      
      
n = z(alpha/2)^2 s^2 / E^2      
      
where      
      
alpha/2 = (1 - confidence level)/2 =    0.025  
      
Using a table/technology,      
      
z(alpha/2) =    1.959963985  
      
Also,      
      
s = sample standard deviation =    12.4  
E = margin of error =    6  
      
Thus,      
      
n =    16.40729745  
      
Rounding up,      
      
n =    17   [ANSWER]

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b)

Note that      
      
n = z(alpha/2)^2 s^2 / E^2      
      
where      
      
alpha/2 = (1 - confidence level)/2 =    0.025  
      
Using a table/technology,      
      
z(alpha/2) =    1.959963985  
      
Also,      
      
s = sample standard deviation =    12.4  
E = margin of error =    3  
      
Thus,      
      
n =    65.62918981  
      
Rounding up,      
      
n =    66   [ANSWER]

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c)

OPTION A: A. Doubling the required accuracy quadruples the sample size. [ANSWER]

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d)

Note that      
      
n = z(alpha/2)^2 s^2 / E^2      
      
where      
      
alpha/2 = (1 - confidence level)/2 =    0.005  
      
Using a table/technology,      
      
z(alpha/2) =    2.575829304  
      
Also,      
      
s = sample standard deviation =    12.4  
E = margin of error =    6  
      
Thus,      
      
n =    28.33838059  
      
Rounding up,      
      
n =    29   [ANSWER]

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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. [ANSWER]

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