Testing Hypotheses
In Exercises 13–24, assume that a simple random sample has been selected and test the given claim. Unless specified by your instructor, use either the P-value method or the critical value method for testing hypotheses. Identify the null and alternative hypotheses, test statistic, P-value (or range of P-values), or critical value(s), and state the final conclusion that addresses the original claim.


Taxi Fares For the first 40 taxi fares (dollars) listed in Data Set 32 “Taxis” from Appendix B, the mean is \$12.035 and the standard deviation is \$8.361. Use a 0.05 significance level to test the claim that the mean cost of a taxicab ride in New York City is less than \$15.00.

Verified step by step guidance

1. Step 1: Identify the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis represents the claim that the population mean is equal to \$15.00, while the alternative hypothesis represents the claim that the population mean is less than \$15.00. Mathematically, H₀: μ = 15 and H₁: μ < 15.
2. Step 2: Determine the test statistic to use. Since the population standard deviation is not provided and the sample size is relatively small (n = 40), use the t-test for a single mean. The formula for the t-test statistic is: $\frac{\mathrm{x̄}-\mu }{\frac{s}{\sqrt{n}}}$, where x̄ is the sample mean, μ is the hypothesized population mean, s is the sample standard deviation, and n is the sample size.
3. Step 3: Calculate the degrees of freedom (df) for the t-distribution. The degrees of freedom are given by df = n - 1, where n is the sample size. In this case, df = 40 - 1 = 39.
4. Step 4: Determine the critical value or P-value. Using a significance level of α = 0.05 and a one-tailed test (since the alternative hypothesis is μ < 15), find the critical t-value from the t-distribution table or calculate the P-value corresponding to the test statistic.
5. Step 5: Compare the test statistic to the critical value or compare the P-value to the significance level. If the test statistic is less than the critical value or if the P-value is less than α, reject the null hypothesis. Otherwise, fail to reject the null hypothesis. Based on this comparison, state the conclusion in the context of the problem: whether there is sufficient evidence to support the claim that the mean cost of a taxicab ride in New York City is less than \$15.00.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating two competing hypotheses: the null hypothesis (H0), which represents a statement of no effect or no difference, and the alternative hypothesis (H1), which indicates the presence of an effect or difference. The goal is to determine whether there is enough evidence in the sample to reject the null hypothesis in favor of the alternative.

P-value

The P-value is a measure that helps determine the strength of the evidence against the null hypothesis. It represents the probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true. A smaller P-value indicates stronger evidence against the null hypothesis, and if it is less than the predetermined significance level (e.g., 0.05), the null hypothesis is rejected.

Significance Level

The significance level, denoted as alpha (α), is the threshold used to decide whether to reject the null hypothesis. It represents the probability of making a Type I error, which occurs when the null hypothesis is incorrectly rejected. Common significance levels are 0.05 and 0.01, meaning there is a 5% or 1% risk of concluding that a difference exists when there is none. In this context, a significance level of 0.05 is used to evaluate the claim about taxi fares.