Question

Insurance companies are interested in knowing the population percent of drivers who always buckle up before riding in a car. When designing a study to determine this population proportion, what is the minimum number of drivers you would need to survey to be 95% confident that the population proportion is estimated to within 0.05?
If it was later determined that it was important to be more than 95% confident and a new survey was commissioned, how would that affect the minimum number you would need to survey? Why?

a. If the confidence level is increased, then the sample size would need to increase because a higher level of confidence increases the margin of error.
b. If the confidence level is increased, then the sample size would need to decrease because increasing the sample size would create an even larger interval.
c. If the confidence level is increased, then the sample size would need to decrease because we would like the proportion of people who buckle up to be around 50%.
d. If the confidence level is increased, then the sample size would not be affected.

Answer 1

Answer:

The value  is  $n = 384$

The correct option is a

Step-by-step explanation:

From the question we are told that

   The margin of error is  E =  0.05

From the question we are told the confidence level is  95% , hence the level of significance is    

      $\alpha = (100 - 95 ) \%$

=>   $\alpha = 0.05$

Generally from the normal distribution table the critical value  of   is  

   $Z_{\frac{\alpha }{2} } = 1.96$

Generally since the sample proportion is not given we will assume it to be

      $\^ p = 0.5$

Generally the sample size is mathematically represented as  

    $n = [\frac{Z_{\frac{\alpha }{2} }}{E} ]^2 * \^ p (1 - \^ p )$

=>   $n = [\frac{ 1.96 }{0.05} ]^2 *0.5 (1 - 0.5)$

=>   $n = 384$

Generally the margin of error is mathematically represented as  

      $E = Z_{\frac{\alpha }{2} } * \frac{\sigma }{\sqrt{n} }$

Generally if the level of confidence increases,  the critical value of  $\frac{\alpha }{2}$  increase and from the equation for margin of error we see the the critical  value varies directly with the margin of error , hence the margin of error  will increase also  

So  If the confidence level is increased, then the sample size would need to increase because a higher level of confidence increases the margin of error.