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


a. Identify the null hypothesis and the alternative hypothesis.

Verified step by step guidance

1. Step 1: Understand the problem. The goal is to test the claim that more than 3/4 (or 75%) of adults rate themselves as above-average drivers. This is a hypothesis testing problem where we compare a sample proportion to a population proportion.
2. Step 2: Define the null hypothesis (H₀) and the alternative hypothesis (Hₐ). The null hypothesis represents the status quo or no effect, while the alternative hypothesis represents the claim being tested. In this case: H₀: p ≤ 0.75 (the proportion of adults who rate themselves as above-average drivers is less than or equal to 75%) and Hₐ: p > 0.75 (the proportion of adults who rate themselves as above-average drivers is greater than 75%).
3. Step 3: Identify the sample proportion (p̂) from the data. The survey indicates that 86% of the 1020 respondents rated themselves as above-average drivers. Thus, p̂ = 0.86.
4. Step 4: Determine the test statistic formula. Since this is a hypothesis test for a population proportion, use the z-test for proportions. The formula is: $z=\frac{p̂-p}{\sqrt{\frac{p(1-p)}{n}}}$, where p is the hypothesized population proportion (0.75), p̂ is the sample proportion (0.86), and n is the sample size (1020).
5. Step 5: Set up the significance level (α) and critical value. Typically, α = 0.05 is used unless otherwise specified. Use the z-distribution table to find the critical value for a one-tailed test at the chosen significance level. Compare the calculated z-value to the critical value to determine whether to reject or fail to reject the null hypothesis.

Key Concepts

Null Hypothesis

The null hypothesis (H0) is a statement that indicates no effect or no difference, serving as a default position in hypothesis testing. In this context, it posits that the proportion of adults who rate themselves as above average drivers is 75% or less, mathematically expressed as H0: p ≤ 0.75. This hypothesis is tested against the alternative hypothesis to determine if there is enough evidence to reject it.

Alternative Hypothesis

The alternative hypothesis (H1) represents the statement that contradicts the null hypothesis, suggesting that there is an effect or a difference. For this scenario, the alternative hypothesis asserts that more than 75% of adults consider themselves above average drivers, formulated as H1: p > 0.75. This hypothesis is what the test aims to support if the evidence is strong enough.

Test Statistic

A test statistic is a standardized value derived from sample data during a hypothesis test, used to determine whether to reject the null hypothesis. It quantifies the difference between the observed sample proportion and the hypothesized population proportion, taking into account sample size. In this case, the test statistic will help assess if the observed proportion of adults rating themselves as above average significantly exceeds 75%.