Question

Nationwide, it is estimated that 40% of service stations have gas storage tanks that leak to some extent. California is trying to lessen that trend. We want to assess the effectiveness of their efforts by seeing if the percentage of service stations whose tanks leak has decreased. To do this, we randomly sample 27 service stations in California and determine whether there is any evidence of leakage. In our sample, only 7 of the station exhibit any leakage. Is there evidence that their efforts have paid off… that stations are showing less leakage?

Answer 1

Answer:

There is not enough evidence that their efforts have paid of. The stations are not showing less leakage.

Step-by-step explanation:

The population proportion of service station that has gas storage tanks that leak to some extent is p = 0.40.

To test whether this proportion has decreased the hypothesis can be defined as:

H₀: The proportion of service stations whose tanks leak has not decreased, i.e. p ≥  0.40.

Hₐ: The proportion of service stations whose tanks leak has decreased, i.e.    p < 0.40.

Given:

n = 27

X = number of service stations whose tanks leak = 7

The sample proportion of service stations whose tanks leak is:

$\hat p =\frac{X}{n} =\frac{7}{27} =0.2593$

Assume that the significance level of the test is α = 5%.

The test statistic is:

$z=\frac{\hat p-p}{\sqrt{ \frac{p(1-p)}{n} }}=\frac{0.2593-0.40}{\sqrt{\frac{0.40(1-0.40}{27}}} =-1.49$

The value of test statistic is -1.49.

Decision rule:

If the p value is less than the significance level the null hypothesis will be rejected and vice-versa.

The p-value of the test statistic is:

$P(Z$

The p-value = 0.0681 > α = 0.05.

Thus, the null hypothesis is not rejected at 5% level of significance.

Conclusion:

As the null hypothesis was not rejected at 5% level of significance, it can be concluded that the proportion of service stations whose tanks leak has not decreased.