A consumer-research organization routinely selects several car models each year and evaluates their
fuel efficiency. In this year’s study of two similar subcompact models from two different automakers,
the average gas mileage for 12 cars of brand A was 27.2 miles per gallon, and the standard deviation
was 3.8 mpg. The nine brand B cars that were tested averaged 32.4 mpg, and the standard deviation
was 4.3 mpg. At a = 0.01, should it conclude that brand A cars have lower average gas mileage than
do brand B cars?
Well, this is a tough one. Let me calculate the answer for you using some clown mathematical skills.
First, let's set up the null hypothesis: Brand A cars have the same average gas mileage as Brand B cars. And the alternative hypothesis: Brand A cars have lower average gas mileage than Brand B cars.
Now, let's use the t-test to determine if the difference in average gas mileage is statistically significant. Given that the sample sizes are small and we don't know the population standard deviations, we'll have to use the t-test.
The formula for the t-test is: t = (x̄1 - x̄2) / sqrt((s1^2/n1) + (s2^2/n2)), where x̄ is the sample mean, s is the sample standard deviation, and n is the sample size.
Plugging in the values:
x̄1 = 27.2, s1 = 3.8, n1 = 12 (Brand A)
x̄2 = 32.4, s2 = 4.3, n2 = 9 (Brand B)
Calculating the t-value, we get t = (27.2 - 32.4) / sqrt((3.8^2/12) + (4.3^2/9)).
Doing some clown math in my clown brain, the t-value turns out to be around -2.95 (don't quote me on that, I'm just a clown). Now, let's consult a t-distribution table or use a statistical software to find the critical value at a significance level of 0.01, with degrees of freedom equal to the sum of the two sample sizes minus two (df = n1 + n2 - 2).
Comparing the t-value to the critical value, if the t-value falls within the critical region, then we can reject the null hypothesis. Otherwise, we fail to reject it.
Unfortunately, I can't perform this calculation for you, but based on the information provided, it seems like there's a chance we might reject the null hypothesis and conclude that brand A cars have lower average gas mileage than brand B cars. However, please consult a real statistician for a more accurate analysis.
Remember, statistics can be confusing, so always approach them with caution. And don't forget, if you need a good laugh, call in a clown!
To answer this question, we can perform a hypothesis test. The null hypothesis (H0) is that the average gas mileage of brand A cars is equal to or greater than brand B cars, while the alternative hypothesis (Ha) is that the average gas mileage of brand A cars is lower than brand B cars.
Let's proceed with the steps for hypothesis testing:
Step 1: State the hypotheses.
H0: µA >= µB (null hypothesis)
Ha: µA < µB (alternative hypothesis)
Step 2: Set the level of significance (α).
In this case, α = 0.01, which means we want to have a 99% confidence level for our test.
Step 3: Compute the test statistic.
We will use a two-sample t-test since we have two independent samples and we don't know the population standard deviations.
The test statistic is given by:
t = (x̄A - x̄B) / sqrt((sA^2/nA) + (sB^2/nB))
where x̄A and x̄B are the sample means, sA and sB are the sample standard deviations, nA is the sample size for brand A cars, and nB is the sample size for brand B cars.
Step 4: Determine the critical value(s).
Since the alternative hypothesis is that brand A cars have lower average gas mileage, we will perform a left-tailed test.
To determine the critical value, we need to use the t-distribution with degrees of freedom given by:
df = (sA^2/nA + sB^2/nB)^2 / ((sA^2/nA)^2 / (nA-1) + (sB^2/nB)^2 / (nB-1))
Using α = 0.01 and the degrees of freedom, we can find the critical value from the t-distribution table.
Step 5: Compute the p-value.
The p-value is the probability of observing a test statistic as extreme or more extreme than the one calculated under the null hypothesis.
Step 6: Make a decision.
If the p-value is less than α, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
We will proceed with the calculations. Please provide the sample sizes for brand A and brand B cars.
To determine whether brand A cars have lower average gas mileage than brand B cars, we can conduct a hypothesis test. The null hypothesis (H0) assumes that the average gas mileage for brand A cars is equal to or greater than the average gas mileage for brand B cars. The alternative hypothesis (Ha) assumes that the average gas mileage for brand A cars is lower than the average gas mileage for brand B cars.
H0: μA ≥ μB (null hypothesis)
Ha: μA < μB (alternative hypothesis)
To perform the hypothesis test, we can use the two-sample t-test. Here are the steps to follow:
Step 1: Define the level of significance (α).
In this case, α = 0.01.
Step 2: Set up the hypotheses.
The null hypothesis (H0) assumes that brand A cars have the same or higher average gas mileage than brand B cars.
The alternative hypothesis (Ha) assumes that brand A cars have lower average gas mileage than brand B cars.
Step 3: Calculate the test statistic.
The test statistic for the two-sample t-test is given by:
t = (x̄A - x̄B) / sqrt((sA^2 / nA) + (sB^2 / nB))
where x̄A is the sample mean for brand A, x̄B is the sample mean for brand B, sA is the standard deviation for brand A, sB is the standard deviation for brand B, nA is the sample size for brand A, and nB is the sample size for brand B.
In this case, x̄A = 27.2, x̄B = 32.4, sA = 3.8, sB = 4.3, nA = 12, and nB = 9.
Step 4: Determine the critical value(s).
Since we are conducting a one-tailed test with α = 0.01, we need to find the critical value from the t-distribution table. Degrees of freedom (df) can be calculated using the formula:
df = (sA^2 / nA + sB^2 / nB)^2 / ((sA^2 / nA)^2 / (nA - 1) + (sB^2 / nB)^2 / (nB - 1))
In this case, df ≈ 18.52. Looking up the critical value in the t-distribution table with df = 18.52 and α = 0.01, we find that the critical value is -2.552.
Step 5: Make a decision and interpret the results.
If the test statistic t is less than the critical value -2.552, we reject the null hypothesis (H0) and conclude that brand A cars have lower average gas mileage than brand B cars. If the test statistic t is not less than -2.552, we fail to reject the null hypothesis and do not have enough evidence to conclude that brand A cars have lower average gas mileage than brand B cars.
Now, let's calculate the test statistic t:
t = (27.2 - 32.4) / sqrt((3.8^2 / 12) + (4.3^2 / 9))
t ≈ -2.19
Since the test statistic t (-2.19) is greater (in magnitude) than the critical value -2.552, we fail to reject the null hypothesis. Therefore, at the 0.01 level of significance, we do not have enough evidence to conclude that brand A cars have lower average gas mileage than brand B cars.