Answer: B. 264
Step-by-step explanation:
Formula to calculate the sample size 'n' , if the prior estimate of the population proportion (p) is available:
[tex]n= p(1-p)(\dfrac{z}{E})^2[/tex]
, where z = Critical z-value corresponds to the given confidence interval
E= margin of error
Let p be the population proportion of clear days.
As per given , we have
Prior sample size : n= 150
Number of clear days in that sample = 117
Prior estimate of the population proportion of clear days = [tex]p=\dfrac{117}{150}[/tex]
E= 0.05
The critical z-value corresponding to 95% confidence interval = z*= 1.95 (By z-table)
Then, the required sample size will be :
[tex]n= \dfrac{117}{150}(1-\dfrac{117}{150})(\dfrac{1.96}{0.05})^2[/tex]
Simplify ,
[tex]n= (0.1716)(39.2)^2[/tex]
[tex]n= 263.687424\approx264[/tex]
Hence, the sample size necessary to construct this interval =264
Thus the correct option is B. 264
