Question

A large distributor of gasoline claims that $60\%$of all drivers stopping at their service stations choose regular unleaded gas and that premium and supreme are each selected $20\%$of the time. To investigate this claim, researchers collected data from a random sample of drivers who put gas in their vehicles at the distributor’s service stations in a large city. The results were as follows.

Carry out a test of the distributor’s claim at the $5\%$significance level.

Short answer

There is sufficient evidence to support the claim that the distribution of type of gas purchased is not what the distributor claims, thus reject $H_{0.}$.

Step-by-step solution

Step 1. Given information

Given:

Step 2. Calculation

Hypotheses:

$H_0:p_{regular}=0.6,p_{premium}=0.2,p_{supreme}=0.2$

$H_a:$At least one of the proportions is incorrect.

Conditions:

SRS: A random sample was used.

Independent: it seems reasonable that there would be more than $400$people who stop at this gas station over a long period of time.

Count: the expected counts are $240,80,80$, all of which are at least $5.$

Chi square test for ${\mu }_1-{\mu }_2,$

$x^2=13.15,P-value=0.0014,df=2$

Thus, $$\begin{gathered} \alpha =0.05 \\ 0.0014<0.05 \end{gathered}$$

Therefore, There is sufficient evidence to support the claim that the distribution of type of gas purchased is not what the distributor claims, thus reject$H_0.$