Answer:
The correct option is D
Step-by-step explanation:
From the question we are told that
The sample size for male car occupant is [tex]n_1 = 2450[/tex]
The number that wear seat belt is [tex]k_1 = \frac{76}{100} * 2450 = 1862[/tex]
The sample size for female car occupant is [tex]n_2 = 2900[/tex]
The number that wear a seat belt is [tex]k_2 = \frac{74}{100} * 2900 = 2146[/tex]
Generally the population proportion is mathematically represented as
[tex]p = \frac{x_1 + x_2 }{n_1 + n_2 }[/tex]
=> [tex]p = \frac{1862 + 2146 }{2450 + 2900 }[/tex]
=> [tex]p = 0.7492[/tex]
The sample proportion for male car occupant is
[tex]\^ p_1 = \frac{k_1}{n_1}[/tex]
=> [tex]\^ p_1 = \frac{1862}{2450}[/tex]
=> [tex]\^ p_1 = 0.76[/tex]
The sample proportion for female car occupant is
[tex]\^ p_2 = \frac{k_2}{n_2}[/tex]
=> [tex]\^ p_2 = \frac{2146}{2900}[/tex]
=> [tex]\^ p_1 = 0.74[/tex]
The null hypothesis is [tex]H_o : p_1 = p_2[/tex]
The alternative hypothesis is [tex]H_a : p_1 \ne p_2[/tex]
Generally the standard error is mathematically represented as
[tex]SE = \sqrt{\frac{p(1 -p )}{ n_1 + n_2} }[/tex]
=> [tex]SE = \sqrt{\frac{0.7492 (1 -0.7492 )}{ 2450 + 2900} }[/tex]
=> [tex]SE = 0.005926[/tex]
Generally the test statistics is mathematically represented as
[tex]z = \frac{\^ p_1 - \^ p_2}{SE}[/tex]
=> [tex]z = \frac{0.76 -0.74}{0.005926}[/tex]
=> [tex]z = 3.37[/tex]
From the z table the area under the normal curve to the right corresponding to 3.37 is
[tex]P(Z > 3.37) = 0.00037584[/tex]
Generally the p-value is mathematically represented as
[tex]p-value = 2 * P(Z > 3.37)= 2* 0.00037584 = 0.0007 5[/tex]
From the value obtained we see that [tex]p-value < \alpha[/tex] hence
The decision rule is
Reject the null hypothesis
The conclusion is
There appears to be a gender gap because there is a significant difference in the proportions
