A company with a large fleet of cars hopes to keep gasoline costs down and sets a goal of attaining a fleet average of at least 26 miles per gallon. To see if the goal is being met they check the gasoline usage for 50 company trips chosen at random, finding a mean of 25.02 mpg and a standard deviation of 4.83 mpg. Is this strong evidence that they have failed to attain their fuel economy goal?

A.) Write appropriate hypothesis

B.) Are there necessary assumptions to make inferences satisfied?

C.) Describe the sampling distribution models of mean fuel economy for samples like this.

D.) Find the P- Value

E.) Explain what the p-vale means in this context?

F.) State an appropriate conclusion.

I'll give a few hints and let you take it from there.

Hypotheses:
Ho: µ ≥ 26
Ha: µ < 26

Try a z-test to determine the test statistic.

Formula:
z = (sample mean - population mean)/(standard deviation divided by the square root of the sample size)

With your data:
z = (25.02 - 26)/(4.83/√50)

Finish the calculation.

Once you have the z-test statistic, check a z-table for the p-value. The p-value is the actual level of the test statistic.

Determine your conclusion. If the null (Ho) is not rejected, there is no difference. If the null is rejected, accept Ha and conclude a difference.

I hope this will help.

t-test is used because it is quantitative, z-test is used for categorical for these type of stats.

Step 1: Hypotheses:

Ho: µ ≥ 26
Ha: µ < 26

Step 2: One sample T- test
t = (ȳ - µ) / (s/√ n)

Step 3: P- Value method
tcdf(p-value, 99,df)

Step 4: reject or not reject null hypothesis

A.) The appropriate hypothesis for this scenario is as follows:

Null Hypothesis (H0): The average fuel economy of the company's fleet is at least 26 miles per gallon.
Alternative Hypothesis (Ha): The average fuel economy of the company's fleet is less than 26 miles per gallon.

B.) In order to make inferences, we need to assume that the gasoline usage for the 50 company trips follows a normal distribution and that the observations are independent of each other.

C.) The sampling distribution model for the mean fuel economy for samples like this can be approximated by a normal distribution. This is based on the Central Limit Theorem, which states that with a sufficiently large sample size, the distribution of the sample mean will be approximately normal.

D.) To find the p-value, we need to calculate the test statistic. Since the sample size is large (n = 50) and the population standard deviation is known (σ = 4.83 mpg), we can use a z-test.

The formula for the test statistic is:
z = (sample mean - hypothesized mean) / (population standard deviation / sqrt(sample size))

In this case, the sample mean is 25.02 mpg, the hypothesized mean is 26 mpg, the population standard deviation is 4.83 mpg, and the sample size is 50.

Calculating the test statistic:
z = (25.02 - 26) / (4.83 / sqrt(50))
z ≈ -1.581

Then, we find the p-value using the z-table or statistical software. The p-value is the probability of observing a test statistic more extreme than the one calculated, assuming the null hypothesis is true.

E.) The p-value represents the probability of obtaining the observed sample mean (or a more extreme value) assuming the null hypothesis is true. In this context, a small p-value would suggest strong evidence against the null hypothesis, indicating that the company has failed to attain its fuel economy goal. A larger p-value would indicate weak evidence against the null hypothesis, suggesting that the company may have achieved its fuel economy goal.

F.) Based on the obtained p-value:

If the p-value is less than the significance level (usually set at 0.05), we reject the null hypothesis in favor of the alternative hypothesis. This means there is strong evidence that the company has failed to attain its fuel economy goal.

If the p-value is greater than the significance level, we fail to reject the null hypothesis. This means there is not enough evidence to suggest that the company has failed to attain its fuel economy goal, and we cannot conclude that they have not met their target.