Question

Two different red-light-running signal systems were installed at various intersection locations with the goal of reducing angle-type crashes. Red-Light-Running System A resulted in 60% angle crashes over a sample of 720 total crashes. Red-Light-Running System B resulted in 52% angle crashes over a sample of 680 total crashes. Was there a difference between the proportions of angle crashes between

Answer 1

Complete Question

Two different red-light-running signal systems were installed at various intersection locations with the goal of reducing angle-type crashes. Red-Light-Running System A resulted in 60% angle crashes over a sample of 720 total crashes. Red-Light-Running System B resulted in 52% angle crashes over a sample of 680 total crashes. Was there a difference between the proportions of angle crashes between the two red-light-running systems installed? Use an alpha of 0.10.

Answer:

Yes there is a difference between the proportions of angle crashes between the two red-light-running systems installed

Step-by-step explanation:

From the question we are told that

   The first sample  proportion  is  $\r p_ 1 = 0.60$

   The  second sample proportion is  $p_2 = 0.52$

    The first sample size is  $n_1 = 720$

     The second sample size is  $n_2 = 680$

     The  level of significance is  $\alpha = 0.10$

     

The null hypothesis is $H_o : \r p_1 - \r p_2 = 0$

The  alternative hypothesis is  $H_a : \r p_1 - \r p_2 \ne 0$

Generally the pooled proportion is mathematically represented as

       $p_p = \frac{(\r p_1 * n_1 ) + (\r p_2 * n_2)}{n_1 + n_2 }$

=>     $p_p = \frac{(0.6 * 720) + ( 0.52 * 680)}{720 +680 }$

=>    $p_p = 0.56$

 Generally the test statistics is evaluated as        

       $t = \frac{ ( \r p_1 - \r p_2 ) - 0 }{ \sqrt{ (p_p (1- p_p) * [ \frac{1}{n_1 } + \frac{1}{n_2 } ])} }$

        $t = \frac{ (0.60 - 0.52 ) - 0 }{ \sqrt{ (0.56 (1- 0.56) * [ \frac{1}{720} + \frac{1}{680 } ])} }$    

       $t = 3.0$

The  p-value obtained from the z-table is  

      $p-value = P(Z> t ) = 0.0013499$

From the question we see that $p-value < \alpha$ so the null hypothesis is rejected

 Hence we can conclude that there is a difference between the proportions