The business college computing center wants to determine the proportion of business students who have laptop computers.If the proportion exceeds 30%, then the lab will scale back a proposed enlargement of its facilities. Suppose 250 business students wererandomly sampled and 75 have laptops. Test if the proportion exceeds 30%. Use α= 0.5
To test if the proportion of business students with laptop computers exceeds 30%, we can use a hypothesis test. We'll set up the null and alternative hypotheses and calculate the test statistic and the p-value.
Let's define the null hypothesis (H0) as the proportion being less than or equal to 30%. The alternative hypothesis (Ha) will be that the proportion exceeds 30%.
H0: Proportion ≤ 0.3
Ha: Proportion > 0.3
Next, we'll calculate the test statistic using the formula:
test statistic = (p̂ - p0) / √((p0 * (1 - p0)) / n)
where:
p̂ is the sample proportion,
p0 is the hypothesized proportion under the null hypothesis,
and n is the sample size.
In this case, p̂ is calculated as the number of students with laptops divided by the total sample size:
p̂ = 75/250 = 0.3
Substituting these values into the formula, we get:
test statistic = (0.3 - 0.3) / √((0.3 * (1 - 0.3)) / 250)
test statistic = 0 / √((0.3 * 0.7) / 250)
test statistic = 0
The p-value is the probability of obtaining a test statistic as extreme as the one calculated, assuming the null hypothesis is true.
We can now compare the test statistic to the critical value corresponding to the significance level of α = 0.05 (given α = 0.5% in the question) and determine if we reject or fail to reject the null hypothesis.
Since the test statistic is 0, which falls within the range of acceptance, the p-value will also be greater than α = 0.05. Thus, we fail to reject the null hypothesis.
Therefore, based on this test, we do not have enough evidence to conclude that the proportion of business students with laptops exceeds 30%.
so, what is α supposed to represent?
75/250 = 30%