Q 8.21. - Elementary Statistics

Question

Fuel Tank Capacity. Consumer Reports provides information on new automobiles models-including price, mileage ratings, engine size, body size, and indicators of features. A simple random sample of $35$ new models yielded the following data on fuel tank capacity, in gallons. 

a. Determine a point estimate for the mean fuel tank capacity of all new automobile models. Interpret your answer in words. (Note: $\Sigma x_i=664.9$ gallons.)

b. Determine a $95\%$ confidence interval for the mean fuel tank capacity of all new automobile models. Assume $\sigma =3.50$ gallons.

c. How would you decide whether fuel tank capacities for new automobile models are approximately normally distributed?

d. Must fuel tank capacities for new automobile models be exactly normally distributed for the confidence interval that you obtained in part (b) to be approximately correct? Explain your answer.

Step-by-Step Solution

Answer

Part (a) The point estimate for the population mean is $19$ gallons.

Part (b) The $95\%$ confidence interval for the mean gasoline tank capacity of all new automotive models is $(17.82,20.18)$

Part (c) It's reasonable to say that fuel tank capacities for new automotive models are roughly evenly dispersed.

Part (d) The confidence interval calculated in component (b) can be considered approximately right.

Part (a) Step 1: Given information

17.2	23.1	17.5	15.7	19.8	16.9	15.3
18.5	18.5	25.5	18.0	17.5	14.5	20.0
17.0	20.0	24.0	26.0	18.1	21.0	19.3
20.0	20.0	12.5	13.2	15.9	14.5	22.2
21.1	14.4	25.0	26.4	16.9	16.4	23.0

Part (a) Step 2: Concept

The formula used: $\overline{x}=\frac{\sum x_i}{n}$ and $\overline{x}\pm 2\frac{\sigma }{\sqrt{n}}$

Part (a) Step 3: Calculation

Get a ballpark figure for the average gasoline tank capacity of all new car models.

From the given information, $\sum x_i=664.9$ gallons and $n=35$

The population mean $\left(\mu \right)$ is estimated using the sample mean $\left(\overline{x}\right)$

The sample mean is,

$$\begin{gathered} \overline{x}=\frac{\sum x_i}{n} \\ =\frac{664.9}{35} \\ =18.99 \\ \approx 19 \end{gathered}$$

Therefore, the point estimate for the population mean is $19$ gallons.

Part (b) Step 1: Calculation

Find the $95\%$ confidence interval for all new car models' average fuel tank capacity.

Assume that $\sigma =3.50$ gallons .

Empirical rule:

Property 1: Around $68\%$ of the data set is located between $(\overline{x}-s,\overline{ x}+s)$

Property 2: Approximately $95\%$ of the data set is located between $(\overline{x}-2s,\overline{ x}+2s)$

Property 3: Approximately $99.7\%$ of the data set is contained within the range $(\overline{x}-3s,\overline{ x}+3s)$

Using Property $2$ $95\%$ of all observations fall within two standard deviations of the mean on either side.

The $95\%$ confidence interval for the population mean is,

$$\begin{gathered} \overline{x}\pm 2\frac{\sigma }{\sqrt{n}}=19\pm 2\left(\frac{3.5}{\sqrt{35}}\right) \\ =19\pm 1.18 \\ =(19-1.18,19+1.18) \\ =(17.82,20.18) \end{gathered}$$

The $95\%$ confidence interval for the mean gasoline tank capacity of all new automotive models is thus $(17.82,20.18)$

Part (c) Step 1: Calculation

Normal distribution conditions:

• Histogram: The distribution has a bell-shaped form (symmetric).
• Probability plot: Every point is getting closer to a straight line.

MINITAB is used to create the normal probability plot.

Procedure for MINITAB:

Step 1: Select Probability Plot from the Graph menu.

Step 2: Click OK after selecting Single.

Step 3: Add the column of fuel tank capacity to the graph variables.

Step 4: Click the OK button.

MINITAB OUTPUT 

Observation: All observations on the probability plot of gasoline tank capacity are closer to a straight line. As a result, it's reasonable to say that fuel tank capacities for new automotive models are roughly evenly dispersed.

Part (d) Step 1: Explanation

Because of the huge sample size, the distribution of the sample mean can be approximated using the central limit theorem. As a result, the gasoline tank capacities of modern automotive models do not need to be evenly distributed.

As a result, the confidence interval calculated in component (b) can be considered approximately right.