Question

Congress regulates corporate fuel economy and sets an annual gas mileage for cars. A company with a large fleet of cars hopes to meet the 2011 goal of 30.2mpg or better for their fleet of cars. To see if the goal is being met, they check the gasoline usage for 50 company trips chosen at random, finding a mean of 32.12mpg and a standard deviation of 4.83mpg. In this strong evidence that they have attained their fuel economy goal? a. Define the parameter and state the hypotheses. b. Define the sampling distribution (mean and standard deviation). c. Perform the test and calculate P-value d. State your conclusion. e. Explain what the p-value means in this context.

Answer 1

Answer:

a) Null hypothesis:$\mu \leq 30.2$  

Alternative hypothesis:$\mu > 30.2$  

b) $X \sim N(\mu=32.12, \sigma=4.83)$

And the distribution for the random sample is given by:

$\bar X \sim N(\mu=32.12, \frac{\sigma}{\sqrt{n}}=\frac{4.83}{\sqrt{50}}=0.683)$

c) $t=\frac{32.12-30.2}{\frac{4.83}{\sqrt{50}}}=2.81$    

$p_v =P(t_{(49)}>2.81)=0.0035$  

d) If we compare the p value and the significance level assumed $\alpha=0.01$ we see that $p_v<\alpha$ so we can conclude that we have enough evidence to reject the null hypothesis, so we can conclude that the true mean is significantly higher than 30.2.  

e) The p-value is the probability of obtaining the observed results of a test, assuming that the null hypothesis is correct. And for this case is a value to accept or reject the null hypothesis.

Step-by-step explanation:

1) Data given and notation  

$\bar X=32.12$ represent the sample mean  

$s=4.83$ represent the sample standard deviation

$n=50$ sample size  

$\mu_o =30.2$ represent the value that we want to test

$\alpha$ represent the significance level for the hypothesis test.  

t would represent the statistic (variable of interest)  

$p_v$ represent the p value for the test (variable of interest)  

a. Define the parameter and state the hypotheses.

We need to conduct a hypothesis in order to check if the mean is higher than 30.2 mpg, the system of hypothesis would be:  

Null hypothesis:$\mu \leq 30.2$  

Alternative hypothesis:$\mu > 30.2$  

If we analyze the size for the sample is > 30 but we don't know the population deviation so is better apply a t test to compare the actual mean to the reference value, and the statistic is given by:  

$t=\frac{\bar X-\mu_o}{\frac{s}{\sqrt{n}}}$  (1)  

t-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".  

b. Define the sampling distribution (mean and standard deviation).

Let X the random variable who represent the variable of interest. And we know that the distribution for X is:

$X \sim N(\mu=32.12, \sigma=4.83)$

And the distribution for the random sample is given by:

$\bar X \sim N(\mu=32.12, \frac{\sigma}{\sqrt{n}}=\frac{4.83}{\sqrt{50}}=0.683)$

c. Perform the test and calculate P-value

Calculate the statistic

We can replace in formula (1) the info given like this:  

$t=\frac{32.12-30.2}{\frac{4.83}{\sqrt{50}}}=2.81$    

P-value

The first step is calculate the degrees of freedom, on this case:  

$df=n-1=50-1=49$  

Since is a one right tailed test the p value would be:  

$p_v =P(t_{(49)}>2.81)=0.0035$  

d. State your conclusion.

If we compare the p value and the significance level assumed $\alpha=0.01$ we see that $p_v<\alpha$ so we can conclude that we have enough evidence to reject the null hypothesis, so we can conclude that the true mean is significantly higher than 30.2.  

e. Explain what the p-value means in this context.

The p-value is the probability of obtaining the observed results of a test, assuming that the null hypothesis is correct. And for this case is a value to accept or reject the null hypothesis.