Answer:
The test statistics is [tex]z = -1.56 [/tex]
The p-value is [tex]p-value = 0.05938 [/tex]
Step-by-step explanation:
From the question we are told
The West side sample size is [tex]n_1 = 578[/tex]
The number of residents on the West side with income below poverty level is [tex]k = 76[/tex]
The East side sample size [tex]n_2=688[/tex]
The number of residents on the East side with income below poverty level is [tex]u = 112[/tex]
The null hypothesis is [tex]H_o : p_1 = p_2[/tex]
The alternative hypothesis is [tex]H_a : p_1 < p_2[/tex]
Generally the sample proportion of West side is
[tex]\^{p} _1 = \frac{k}{n_1}[/tex]
=> [tex]\^{p} _1 = \frac{76}{578}[/tex]
=> [tex]\^{p} _1 = 0.1315 [/tex]
Generally the sample proportion of West side is
[tex]\^{p} _2 = \frac{u}{n_2}[/tex]
=> [tex]\^{p} _2 = \frac{112}{688}[/tex]
=> [tex]\^{p} _2 = 0.1628 [/tex]
Generally the pooled sample proportion is mathematically represented as
[tex]p = \frac{k + u}{ n_1 + n_2 }[/tex]
=> [tex]p = \frac{76 + 112}{ 578 + 688 }[/tex]
=> [tex]p =0.1485[/tex]
Generally the test statistics is mathematically represented as
[tex]z = \frac{\^ {p}_1 - \^{p}_2}{\sqrt{p(1- p) [\frac{1}{n_1 } + \frac{1}{n_2} ]} }[/tex]
=> [tex]z = \frac{ 0.1315 - 0.1628 }{\sqrt{0.1485(1-0.1485) [\frac{1}{578} + \frac{1}{688} ]} }[/tex]
=> [tex]z = -1.56 [/tex]
Generally the p-value is mathematically represented as
[tex]p-value = P(z < -1.56 )[/tex]
From z-table
[tex]P(z < -1.56 ) = 0.05938[/tex]
So
[tex]p-value = 0.05938 [/tex]
