The U.S. Energy Administration reported that the mean price...

The U.S. Energy Administration reported that the mean price for a gallon of regular gasoline in the United States was $$\$2.357$$ (U.S. Energy Administration, January 30, 2006). Data for a sample of regular gasoline prices at 50 service stations in the Lower Atlantic states are contained in the data file named Gasoline. Conduct a bypothesis test to determine whether the mean price for a gallon of gasoline in the Lower Adlantic states is different from the national mean. Use $\alpha=.05$ for the level of significance, and state your conclusion.

Key Concepts

Significance Level and p-Value

The significance level, denoted by ?, represents the probability threshold for making a Type I error—rejecting a true null hypothesis. The p-value is the probability of obtaining test results at least as extreme as those observed, assuming the null hypothesis is true. If the p-value is less than or equal to the significance level, the null hypothesis is rejected, indicating statistically significant evidence in favor of the alternative hypothesis.

One-Sample t-Test

When the population standard deviation is unknown and the sample size is moderately small or the data can be assumed to come from a normally distributed population, the one?sample t-test is used to compare the sample mean to a known value. This test utilizes the t-distribution to account for additional uncertainty introduced by estimating the standard deviation from the sample.

Two-Tailed Test

A two-tailed test is used when the alternative hypothesis is concerned with differences in either direction from the hypothesized parameter value. This means that the rejection regions for the test are located in both tails of the test statistic’s sampling distribution, ensuring that any significant deviation, whether greater or smaller than the hypothesized value, is considered.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences or draw conclusions about a population parameter based on sample data. It involves setting up a null hypothesis (a statement of no effect or difference) and an alternative hypothesis (what the researcher aims to support), and then using sample data to determine whether there is sufficient evidence to reject the null hypothesis.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H?) typically states that there is no difference or effect—in this case, that the population mean equals a specific value—while the alternative hypothesis (H? or Ha) states that the parameter of interest is different from that value. This formulation is key for conducting tests that compare a sample statistic against a known or hypothesized population parameter.

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