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 [tex]\r p_ 1 = 0.60[/tex]
The second sample proportion is [tex]p_2 = 0.52[/tex]
The first sample size is [tex]n_1 = 720[/tex]
The second sample size is [tex]n_2 = 680[/tex]
The level of significance is [tex]\alpha = 0.10[/tex]
The null hypothesis is [tex]H_o : \r p_1 - \r p_2 = 0[/tex]
The alternative hypothesis is [tex]H_a : \r p_1 - \r p_2 \ne 0[/tex]
Generally the pooled proportion is mathematically represented as
[tex]p_p = \frac{(\r p_1 * n_1 ) + (\r p_2 * n_2)}{n_1 + n_2 }[/tex]
=> [tex]p_p = \frac{(0.6 * 720) + ( 0.52 * 680)}{720 +680 }[/tex]
=> [tex]p_p = 0.56[/tex]
Generally the test statistics is evaluated as
[tex]t = \frac{ ( \r p_1 - \r p_2 ) - 0 }{ \sqrt{ (p_p (1- p_p) * [ \frac{1}{n_1 } + \frac{1}{n_2 } ])} }[/tex]
[tex]t = \frac{ (0.60 - 0.52 ) - 0 }{ \sqrt{ (0.56 (1- 0.56) * [ \frac{1}{720} + \frac{1}{680 } ])} }[/tex]
[tex]t = 3.0[/tex]
The p-value obtained from the z-table is
[tex]p-value = P(Z> t ) = 0.0013499[/tex]
From the question we see that [tex]p-value < \alpha[/tex] so the null hypothesis is rejected
Hence we can conclude that there is a difference between the proportions
