Question

4. (16 points) A company with a fleet of cars found that in a random sample of 65 chosen and tested cars that 30 had emissions systems which failed to meet pollution control guidelines. A statistician is interested in testing if there is enough evidence to conclude that more than 30% of the fleet might be out of compliance regarding pollution control guidelines. Perform a five step hypothesis test using a 0.05 significance level. Write out the hypotheses: Determine critical value(s): Compute the test statistic: Determine the decision: Write the concluding statement:

Answer 1

Answer:

Null hypothesis:$p\leq 0.3$  

Alternative hypothesis:$p > 0.3$  

Critical value: $z_{\alpha/2}= 1.64$

Test statistic: $z=\frac{0.462 -0.3}{\sqrt{\frac{0.3(1-0.3)}{65}}}=2.85$  

For this case the calculated value is higher than the critical value so then we can reject the null hypothesis. And we can conclude that the true proportion of cars with emissions systems which failed to meet pollution control guidelines for this case is significantly higher than 0.30 or 30%

Step-by-step explanation:

Information given

n=65 represent the random sample taken

X=30 represent the number of cars with emissions systems which failed to meet pollution control guidelines

$\hat p=\frac{30}{65}=0.462$ estimated proportion of cars with emissions systems which failed to meet pollution control guidelines

$p_o=0.30$ is the value to verify

$\alpha=0.05$ represent the significance level

z would represent the statistic

$p_v$ represent the p value

Hypothesis to test

We want to verify if the true proportion for this case is higher than 0.3 or no, the system of hypothesis are:  

Null hypothesis:$p\leq 0.3$  

Alternative hypothesis:$p > 0.3$  

The statistic for this case would be:

$z=\frac{\hat p -p_o}{\sqrt{\frac{p_o (1-p_o)}{n}}}$ (1)  

Replacing the info we got:

$z=\frac{0.462 -0.3}{\sqrt{\frac{0.3(1-0.3)}{65}}}=2.85$  

The critical value for this case would be taking in count the significance level in the right tail:

$z_{\alpha/2}= 1.64$

For this case the calculated value is higher than the critical value so then we can reject the null hypothesis. And we can conclude that the true proportion of cars with emissions systems which failed to meet pollution control guidelines for this case is significantly higher than 0.30 or 30%