Question

A resident of Bayport claims to the City Council that the proportion of Westside residents

(1) with income below the poverty level is lower than the proportion of Eastside residents
(2) The City Council decides to test this claim by collecting a random sample of resident incomes from the Westside of town and a random sample of resident incomes from the Eastside of town. Seventy-six out of 578 Westside residents had an income below the poverty level. Hundred-and-twelve out of 688 Eastside residents had an income below the poverty Specify the hypotheses.
Calculate the value of the test statistic (round to 4 decimal places).
Calculate the p-value (round to 4 decimal places).

Answer 1

The test statistics is
`z =  -1.56`

The p-value is
`p-value =  0.05938`

Explanation:

From the question we are told

The West side sample size is
`n_1  =  578`

The number of residents on the West side with income below poverty level is
`k  = 76`

The East side sample size
`n_2=688`

The number of residents on the East side with income below poverty level is
`u  = 112`

The null hypothesis is
`H_o  :  p_1 = p_2`

The alternative hypothesis is
`H_a :  p_1 <  p_2`

Generally the sample proportion of West side is

`\^(p) _1 = (k)/(n_1)`

=>
`\^(p) _1 = (76)/(578)`

=>
`\^(p) _1 =  0.1315`

Generally the sample proportion of West side is

`\^(p) _2 = (u)/(n_2)`

=>
`\^(p) _2 = (112)/(688)`

=>
`\^(p) _2 =  0.1628`

Generally the pooled sample proportion is mathematically represented as

`p = (k + u)/( n_1 + n_2 )`

=>
`p = (76 + 112)/( 578 + 688 )`

=>
`p =0.1485`

Generally the test statistics is mathematically represented as

`z = \frac{\^ {p}_1 - \^(p)_2}{\sqrt{p(1- p) [(1)/(n_1 ) + (1)/(n_2)  ]}  }`

=>
`z = \frac{ 0.1315  - 0.1628 }{\sqrt{0.1485(1-0.1485) [(1)/(578) + (1)/(688)  ]}  }`

=>
`z =  -1.56`

Generally the p-value is mathematically represented as

`p-value =  P(z <  -1.56 )`

From z-table

`P(z <  -1.56 ) =  0.05938`

So

`p-value =  0.05938`