Operators of gasoline-fueled vehicles complain about the...

Operators of gasoline-fueled vehicles complain about the price of gasoline in gas stations. According to the American Petroleum Institute, the federal gas tax per gallon is constant $(18.4 c$ as of January 13,2005 ), but state and local taxes vary from $7.5 c$ to $32.10 c$ for $n=18$ key metropolitan areas around the country. $^{\star}$ The total tax per gallon for gasoline at each of these 18 locations is given next. Suppose that these measurements constitute a random sample of size 18 $$\begin{aligned} &\begin{array}{llllll} 42.89 & 53.91 & 48.55 & 47.90 & 47.73 & 46.61 \end{array}\\ &\begin{array}{llllll} 40.45 & 39.65 & 38.65 & 37.95 & 36.80 & 35.95 \end{array}\\ &\begin{array}{llllll} 35.09 & 35.04 & 34.95 & 33.45 & 28.99 & 27.45 \end{array} \end{aligned}$$ a. Is there sufficient evidence to claim that the average per gallon gas tax is less than $45 \%$ ? Use the $t$ table in the appendix to bound the $p$ -value associated with the test. b. What is the exact $p$ -value?: c. Construct a $95 \%$ confidence interval for the average per gallon gas tax in the United States.

Operators of gasoline-fueled vehicles complain about the price of gasoline in gas stations. According to the American Petroleum Institute, the federal gas tax per gallon is constant $(18.4 c$ as of January 13,2005 ), but state and local taxes vary from $7.5 c$ to $32.10 c$ for $n=18$ key metropolitan areas around the country. $^{\star}$ The total tax per gallon for gasoline at each of these 18 locations is given next. Suppose that these measurements constitute a random sample of size 18 $$\begin{aligned}
&\begin{array}{llllll}
42.89 & 53.91 & 48.55 & 47.90 & 47.73 & 46.61
\end{array}\\
&\begin{array}{llllll}
40.45 & 39.65 & 38.65 & 37.95 & 36.80 & 35.95
\end{array}\\
&\begin{array}{llllll}
35.09 & 35.04 & 34.95 & 33.45 & 28.99 & 27.45
\end{array}
\end{aligned}$$ a. Is there sufficient evidence to claim that the average per gallon gas tax is less than $45 \%$ ? Use the $t$ table in the appendix to bound the $p$ -value associated with the test. b. What is the exact $p$ -value?: c. Construct a $95 \%$ confidence interval for the average per gallon gas tax in the United States.

Key Concepts

One-Sample t-Test

The one-sample t-test is a statistical procedure used to determine whether the mean of a single sample is significantly different from a known or hypothesized population mean. This test is appropriate when the population standard deviation is unknown and the sample size is relatively small, leading to reliance on the t-distribution rather than the normal distribution.

p-Value

The p-value is the probability, under the assumption that the null hypothesis is true, of obtaining a result equal to or more extreme than what was actually observed. It is used to assess the strength of the evidence against the null hypothesis; a small p-value indicates strong evidence that the null hypothesis may be false.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter with a specified level of confidence (commonly 95%). It provides an estimate of the uncertainty around the sample estimate and a sense of the range within which the true population mean is likely to fall.

Degrees of Freedom

Degrees of freedom in the context of the t-test refer to the number of independent values that can vary in the analysis after certain restrictions are imposed, such as estimating parameters like the sample mean. In a one-sample t-test, the degrees of freedom are typically equal to the sample size minus one, and they affect the exact shape of the t-distribution used for hypothesis testing.

Transcript

00:01 So in this problem, we're looking at the total tax per gallon for gasoline.

00:07 And we are going to look at a sample of 18 different gas stations.

00:14 And the question here is that is there sufficient evidence to claim that the average per gallon gas tax is less than 45 cents? so we are trying to see if the average is less than 45 cents.

00:38 So the first thing we are going to have to do, since we are running this test, is to write our null hypothesis and our alternative hypothesis.

00:49 And a null hypothesis is always going to be a statement of equality.

00:53 And our alternative hypothesis is a statement of inequality.

00:58 Since we are trying to test something that's of inequality, our alternative is going to be that mu is less than 45.

01:09 Now the complementary null hypothesis would be that mu is equal to 45.

01:18 So because our alternative hypothesis has a less than symbol, this is going to end up being a left -tailed test.

01:32 And we are going to need a test statistic.

01:37 And to calculate our test statistic, we are going to have to assume normal distribution.

01:44 And we will apply the formula, y bar minus mu, divided by s over the square root of n.

01:55 So we're going to have to take this set of data and find the average of that set of data as well as the standard deviation.

02:05 So the most efficient way of doing that is to bring in our calculator.

02:11 We're going to hit the stat feature and edit.

02:15 And as you can see, i have already put the 18 pieces of data into the calculator.

02:21 So we're going to do stat.

02:23 We're going to scoot over to the calculate menu.

02:28 We're going to do one variable statistics of everything that we have in list one.

02:33 And we are going to find that the average is approximately 39 .556.

02:40 And you can see our standard deviation is going to be 7 .137.

02:49 So we're going to write this down.

02:50 So we said our y bar is about 39.

02:55 Point 556 and our standard deviation will be 7 .13764184.

03:08 So we're now ready to utilize our formula and find our standardized test statistic.

03:15 So we're going to have 39 .556 minus 45 all over the sample standard deviation divided by the square root of our sample size.

03:34 And in doing so, we are going to get a t score of a negative 3 .236.

03:43 So now that we have set up our information, we're ready to begin answering the questions.

03:52 So ultimately, we've got to answer question a.

03:58 So is there sufficient evidence to claim that the average per gallon gas tax is less than 45 cents? and we want to use the table in the appendix to bound the p value associated with this test.

04:13 So ultimately we'll answer the test in the next section.

04:19 So we want to start with bounding.

04:22 So we are trying to find the p value.

04:28 And that p value is going to be the smallest value of alpha or the level of significance for which the null hypothesis can be rejected.

04:56 And in order to bound this, we are going to have to take a look at the t distribution in the back of your textbook.

05:06 And if you look across the top of that.

05:11 Chart, you're going to see t values for areas of 0 .10, a t value 4 .050, a t value 4 .025, a t value 4 .020, a t value 4 .010, and a t value 4 .005.

05:35 And we need to focus on the row that has the degrees of freedom of 17.

05:43 Now degrees of freedom, since we are working with a t distribution, which is a family of curves, the degrees of freedom gives us an indication of what the shape of the graph looks like.

05:55 And our degrees of freedom will always be found by doing the sample size minus 1.

06:02 So our sample size was 18.

06:04 So we're going to do 18 minus 1, and we're going to focus on the row has a degree of freedom of 17.

06:14 And when you look across the row, you're going to see these values.

06:32 Now, because this is a left -tailed test, the values in this chart are basically the t scores for on the right side of the curve.

06:44 But since it's a t distribution and it is symmetric with a center of zero, these values would also apply to the left side, but they would all be negative.

07:00 So we're going to throw some negatives in here.

07:05 And we calculated our t or test statistic to be negative 3 .296, which ends up kind of like right in here.

07:19 So technically we could say that t, which was negative 3 .236, is less than negative 2 .898, but is greater than negative negative infinity or negative infinity.

07:40 So if we look at the values that go with the negative 2 .98, we have an area or a probability of 0 .005.

07:51 So our p value is going to end up being less than 0 .005, but greater than 0 .0.

08:05 So that would be the bound of the p value associated with this particular test.

08:13 Now as we go into part b, in part b, we are asked to find the exact p value.

08:21 So in order to find the exact p value, we're going to have to use their applet or we could use the graphing calculator again.

08:35 So i'm going to show you both ways...

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