Question

In a recent year, 6% of cars sold had a manual transmission. A random sample of college students who owned cars revealed the following: out of 124 cars, 29 had manual transmissions. Estimate the proportion of college students who drive cars with manual transmissions with 90% confidence.

Answer 1

Answer:

The  estimate the proportion of college students who drive cars with manual transmissions with 90% confidence is  

      $0.171 <  p <  0.297$  

Step-by-step explanation:

From the question we are told that

    The sample size is  n  =  124

    The population proportion p =  0.06

     The number of cars with manual transmissions is  k =  29

From the question we are told the confidence level is  90% , hence the level of significance is    

      $\alpha = (100 - 90 ) \%$

=>   $\alpha = 0.10$

Generally the sample proportion is mathematically represented as

       $\^ p = \frac{ 29 }{124 }$

=>   $\^ p = 0.234$

Generally from the normal distribution table the critical value  of  $\frac{\alpha }{2}$ is  

   $Z_{\frac{\alpha }{2} } =  1.645$  

Generally the margin of error is mathematically represented as  

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

=>   $E =   1.645  * \sqrt{\frac{0.234 (1- 0.234 )}{124} }$

=>   $E =  0.063  $

Generally 95% confidence interval is mathematically represented as  

      $\^ p -E <  p <  \^ p +E$

=>  $0.234  - 0.063  <  p <  0.234  + 0.063$

=>  $0.171 <  p <  0.297$