Statistical Literacy and Critical Thinking
In Exercises 1–4, use the results from a Hankook Tire Gauge Index survey of a simple random sample of 1020 adults. Among the 1020 respondents, 86% rated themselves as above average drivers. We want to test the claim that more than 3/4 of adults rate themselves as above average drivers.


Null and Alternative Hypotheses and Test Statistic


b. Find the value of the test statistic.

Verified step by step guidance

1. Step 1: Define the null and alternative hypotheses. The null hypothesis (H₀) states that the proportion of adults who rate themselves as above-average drivers is less than or equal to 3/4 (p ≤ 0.75). The alternative hypothesis (H₁) states that the proportion is greater than 3/4 (p > 0.75).
2. Step 2: Identify the sample proportion (p̂) and sample size (n). From the problem, 86% of the 1020 respondents rated themselves as above-average drivers. Thus, p̂ = 0.86 and n = 1020.
3. Step 3: Calculate the standard error (SE) of the sample proportion using the formula: SE = √((p₀ * (1 - p₀)) / n), where p₀ is the hypothesized proportion (0.75).
4. Step 4: Compute the test statistic (z) using the formula: z = (p̂ - p₀) / SE. Substitute the values of p̂, p₀, and SE into the formula to find the z-score.
5. Step 5: Interpret the test statistic. Compare the calculated z-score to the critical value for the chosen significance level (e.g., α = 0.05) or use the p-value approach to determine whether to reject or fail to reject the null hypothesis.

Key Concepts

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H0) represents a statement of no effect or no difference, while the alternative hypothesis (H1) indicates the presence of an effect or difference. In this context, the null hypothesis would state that 75% or fewer adults rate themselves as above average drivers, while the alternative hypothesis would claim that more than 75% do. Understanding these hypotheses is crucial for determining the direction and purpose of the statistical test.

Test Statistic

A test statistic is a standardized value that is calculated from sample data during a hypothesis test. It measures how far the sample statistic is from the null hypothesis value, expressed in terms of standard errors. In this case, the test statistic will help determine whether the observed proportion of adults rating themselves as above average significantly exceeds the hypothesized proportion of 75%, allowing for a decision regarding the null hypothesis.

P-value

The p-value is a probability that measures the strength of the evidence against the null hypothesis. It indicates the likelihood of observing the sample data, or something more extreme, if the null hypothesis is true. A low p-value (typically less than 0.05) suggests that the observed data is unlikely under the null hypothesis, leading to its rejection in favor of the alternative hypothesis. Understanding the p-value is essential for interpreting the results of the hypothesis test.