Answer:
The sample size is [tex]n = 600[/tex]
Step-by-step explanation:
From the question we are told that
The sample proportion is [tex]\r p = 0.48[/tex]
The margin of error is [tex]MOE = 0.04[/tex]
Given that the confidence level is 95% the level of significance is mathematically represented as
[tex]\alpha = 100 - 95[/tex]
[tex]\alpha = 5 \%[/tex]
[tex]\alpha = 0.05[/tex]
Next we obtain the critical value of [tex]\frac{\alpha }{2}[/tex] from the normal distribution table , the values is
[tex]Z_{\frac{\alpha }{2} } = 1.96[/tex]
The reason we are obtaining critical value of [tex]\frac{\alpha }{2}[/tex] instead of [tex]\alpha[/tex] is because
[tex]\alpha[/tex] represents the area under the normal curve where the confidence level interval ( [tex]1-\alpha[/tex]) did not cover which include both the left and right tail while
[tex]\frac{\alpha }{2}[/tex] is just the area of one tail which what we required to calculate the margin of error
Generally the margin of error is mathematically represented as
[tex]MOE = Z_{\frac{\alpha }{2} } * \sqrt{ \frac{\r p(1- \r p )}{n} }[/tex]
substituting values
[tex]0.04= 1.96* \sqrt{ \frac{0.48(1- 0.48 )}{n} }[/tex]
[tex]0.02041 = \sqrt{ \frac{0.48(52 )}{n} }[/tex]
[tex]0.02041 = \sqrt{ \frac{ 0.2496}{n} }[/tex]
[tex]0.02041^2 = \frac{ 0.2496}{n}[/tex]
[tex]0.0004166 = \frac{ 0.2496}{n}[/tex]
=> [tex]n = 600[/tex]
