Question

Better parking A local high school makes a change that should improve student

satisfaction with the parking situation. Before the change, $37\%$of the school’s students approved of the parking that was provided. After the change, the principal surveys an SRS of $200$from the more than $2500$students at the school. In all, $83$students say that they approve of the new parking arrangement. The principal cites this as evidence that the change was effective.

a. Describe a Type I error and a Type II error in this setting, and give a possible

consequence of each.

b. Is there convincing evidence that the principal’s claim is true?

Short answer

a. The change is not effective. When effective, it won't improve parking.

b. No evidence that principal's claim is true.

Step-by-step solution

Given Information

It is given that claim is greater than $37\%$.

$\alpha =0.05$

$n=200$

$x=83$

Type I and Type II Error

The claim is null or alternate hypothesis.

Null hypothesis: $H_0:p=37\%=0.37$

Alternate Hypothesis: $H_1:p>0.37$

Once null hypothesis is true, type I error rejects the null hypothesis:

Evidence is present that students who approve of new parking is $>0.37$, when students who approve parking is actually $0.37$

Conclusion is change was effective when it was actually not effective and it may lead to wastage of money.

Once null hypothesis is false, type II error fails to reject it:

No evidence is present that students who approve of new parking is really larger than $0.37$.

Change is not effective. When it was, parking is not improved.

Checking if Principal's claim is true or not.

The condition of normality is satisfied as:

$n{p}_0=200(0.37)=74$and $n\left(1-p_0\right)=200(1-0.37)=126$. Both are large than $10$.

Conditions are satisfied, we can use hypothesis test.

Sample proportion is $\hat{p}=\frac{x}{n}=\frac{83}{200}=0.415$

Test static: $z=\frac{\hat{p}-p_0}{\sqrt{\frac{p_0\left(1-p_0\right)}{n}}}=\frac{0.415-0.37}{\sqrt{\frac{0.37(1-0.37)}{200}}}=1.32$

$P$value is $P=P(z>1.32)$

$=1-P(Z<1.32)=1-0.9066=0.0934$

Hence, $P>0.05\Rightarrow \text{Fail to reject}{H}_0$.

No evidence that principal's claim is true.