Question

The business college computing center wants to determine the proportion of business students who have personal computers (PC's) at home. If the proportion exceeds 30%, then the lab will scale back a proposed enlargement of its facilities. Suppose 300 business students were randomly sampled and 65 have PC's at home. What assumptions are necessary for this test to be satisfied

Answer 1

Given data :

x = 65, n = 300

`$\hat p = (x)/(n)`

`$=(65)/(300)$`

= 0.2167

The hypothesis are :

`$H_0: p \leq 0.3$`

`$H_0: p> 0.3$`

The
`\text{level of significance}`, α = 0.05

The test is right tailed.

The standard deviation is :

`$\sigma = \sqrt{(0.3(1-0.3))/(300)}$`

σ = 0.0265

The test statistics is :

`$z=(\hat p - p)/(\sigma)$`

`$z=(0.2167 - 0.3)/(0.0265)$`

= -3.14

The critical value is 1.645

The rejection region is : If z > 1.645, then we reject
`H_0`

Decision :

Since the test statistics does not lie in the rejection, so we fail to .
`\text{reject the null hypothesis}`.

P-value :
`$P(z > - 3.14)$` = 0.9992

Therefore, the p-value is not less than the level of significance so we fail to
`\text{reject the null hypothesis}`.