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

Solution :

Given data :

x = 65, n = 300

$\hat p = \frac{x}{n}$

  $=\frac{65}{300}$

  = 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 :

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

σ = 0.0265

The test statistics is :

$z=\frac{\hat p - p}{\sigma}$

$z=\frac{0.2167 - 0.3}{0.0265}$

  = -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 $\text{reject the null hypothesis}$.