Question

In an attempt to reduce their carbon footprint, many consumers are purchasing hybrid, plug-in hybrid, or electric cars. Consumer Reports ranks the Chevrolet Bolt first among electric cars, with an EPA rating of 238 miles between battery charges, although others report a range between 190 and 313 miles! \(!^{10}\) To test this claim, suppose that \(n=35\) road tests were conducted and that the average miles between charges was 232 miles with a standard deviation of 20.2 miles. a. Construct a \(95 \%\) confidence interval for \(\mu,\) the average time between battery charges for the Chevrolet Bolt. b. Does the confidence interval in part a confirm the EPA's claim of 238 miles per battery charge? Why or why not?

Short answer

Based on the analysis and solution, the short answer could be formed as follows: A 95% confidence interval was constructed to find the mean miles between charges for the Chevrolet Bolt, using a sample size of 35, a sample mean of 232, and a sample standard deviation of 20.2. The result was a 95% confidence interval of approximately 225.08 to 238.92 miles. When evaluating the EPA's claim of 238 miles per charge, it was found to fall within this confidence interval. Thus, while the data does not explicitly confirm the EPA's claim, it also doesn't contradict it. This leads to the conclusion that, according to the data collected, the EPA's claim is plausible. However, more data would be necessary to confirm this claim with 95% certainty.

Step-by-step solution

Calculate the Standard Error

The standard error of the sample mean (SE) is calculated by dividing the sample standard deviation (s) by the square root of the sample size (n).$$SE = \frac{s}{\sqrt{n}}$$Substitute the values.$$SE = \frac{20.2}{\sqrt{35}} \approx 3.41$$ The standard error is approximately 3.41 miles.

Determine the t-score

A t-score is necessary to construct a 95% confidence interval because the population standard deviation is unknown. Since the desired confidence level is 95% and the degrees of freedom (df) for the sample are 34 (n-1), we can find the t-score using a t-table or other statistical resources. For a 95% confidence level and 34 degrees of freedom, the t-score is approximately 2.032.

Construct the Confidence Interval

Now that we have the t-score and standard error, we can construct the 95% confidence interval for the population mean \(\mu\). The formula for the confidence interval is: $$\bar{x} \pm t \times SE$$ Substitute the values and calculate the interval. $$232 \pm 2.032 \times 3.41 \approx 232 \pm 6.92$$ So, the 95% confidence interval for the average miles between charges for the Chevrolet Bolt is approximately \((225.08, 238.92)\) miles.

Evaluate EPA's Claim

Now, we need to determine if the EPA's claim of 238 miles per charge falls within the calculated confidence interval. Since the EPA's claim of 238 miles falls within the interval \((225.08, 238.92)\), we can say that the data from the 35 road tests neither supports nor negates the EPA's claim. The claim is plausible, but we cannot confirm it with the given data at a 95% confidence level.

Key concepts

Standard Error

When conducting statistics involving sample data, one key concept you'll encounter is the standard error (SE). It measures the accuracy with which a sample represents a population. In simpler terms, it tells us how far we might expect the sample mean to deviate from the actual population mean.

The formula for standard error is: \[\begin{equation}SE = \frac{s}{\sqrt{n}}\end{equation}\]where s is the sample standard deviation and n is the sample size. The SE decreases as the sample size increases, because larger samples tend to be more representative of the population. This relationship is crucial when interpreting the results of a study because it gives context to how precise our estimates are. Imagine shooting arrows at a target; standard error is akin to measuring how closely your arrows cluster together. A smaller cluster (low SE) means you're pretty consistent—your shots are accurate representations of your aiming point.

T-Score

In the realm of statistics, the t-score is especially important when dealing with smaller sample sizes or when the population standard deviation is unknown. A t-score, sometimes called a t-value, helps to determine how far a sample statistic (like the mean) deviates from the assumed population parameter, standardized by the standard error.

The t-score is calculated using the formula:\[\begin{equation}t = \frac{\bar{x} - \mu}{SE}\end{equation}\]where \begin{itemize}

\bar{x} is the sample mean,

\mu is the population mean, and

SE is the standard error.

Finding the correct t-score for a given confidence interval involves degrees of freedom (df), which in a sample is typically n-1. Using a t-distribution table, you can match the desired confidence level and df to find the t-score that will create interval estimates for the population mean. Comparing this to an archery metaphor, if standard error is the cluster tightness, the t-score tells you how far the center of your cluster is from the bullseye.

Population Mean

The population mean (\[\begin{equation}\mu\end{equation}\]), often the parameter of interest in many research endeavors, represents the average of a characteristic for an entire population. In studies, we frequently use sample data to estimate this value because it's usually impractical or impossible to measure every individual in the population.

However, because we're working with samples, our estimate — the sample mean (\[\begin{equation}\bar{x}\end{equation}\]) — will have some level of uncertainty associated with it. This is where confidence intervals come in: they provide a range of values that likely include the population mean. The population mean is like the hidden treasure, and our statistical methods are the map; we can't find the exact spot where 'X marks the spot,' but we can draw a circle where we're pretty sure the treasure lies within.

Hypothesis Testing

Finally, let's delve into hypothesis testing, a cornerstone of statistical inference. Essentially, hypothesis testing is a structured process used to determine if there is enough evidence to support a certain belief (hypothesis) about a population parameter.

The process begins with stating two opposing hypotheses: the null hypothesis (\[\begin{equation}H_{0}\end{equation}\]) and the alternative hypothesis (\[\begin{equation}H_{a}\end{equation}\]). The null hypothesis represents the status quo or a skeptical perspective, while the alternative represents what you're attempting to provide evidence for. In our vehicle example, we can consider the null hypothesis to be that the true average mileage is 238 miles, as the EPA claims. We then collect sample data and calculate test statistics, which are compared to a pre-determined threshold to decide whether we should reject the null hypothesis or not. If the data are unlikely under the null hypothesis, we reject it in favor of the alternative. Think of it like a court trial for the null hypothesis; unless the evidence strongly suggests otherwise, the null hypothesis will stand.