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.


Number and Proportions


b. Identify the sample proportion and use the symbol that represents it.

Verified step by step guidance

1. Step 1: Understand the problem. The goal is to identify the sample proportion from the given data and use the appropriate statistical symbol to represent it. The sample proportion is the ratio of respondents who rated themselves as above average drivers to the total number of respondents.
2. Step 2: Recall the formula for sample proportion. The sample proportion (denoted as $p$_̂) is calculated as: $p$_̂=xn, where $x$ is the number of respondents who rated themselves as above average drivers, and $n$ is the total sample size.
3. Step 3: Extract the values from the problem. The total sample size is $n=1020$, and the percentage of respondents who rated themselves as above average drivers is $86\%$. Convert this percentage to a proportion by dividing by 100: $0.86$.
4. Step 4: Calculate the number of respondents who rated themselves as above average drivers. Multiply the proportion ($0.86$) by the total sample size ($1020$): $x=0.86\times 1020$.
5. Step 5: Substitute the values into the formula for sample proportion. Use $p$_̂=xn to represent the sample proportion, where $x$ is the number of respondents calculated in Step 4 and $n$ is $1020$.

Key Concepts

Sample Proportion

The sample proportion is a statistic that represents the fraction of a sample that possesses a certain characteristic. It is calculated by dividing the number of individuals in the sample with the characteristic by the total number of individuals in the sample. In this case, the sample proportion of adults who rate themselves as above average drivers can be expressed as 0.86, or 86%, based on the survey results.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about a population based on sample data. It involves formulating a null hypothesis (the default assumption) and an alternative hypothesis (the claim being tested). In this scenario, the null hypothesis would state that 75% or fewer adults rate themselves as above average drivers, while the alternative hypothesis would assert that more than 75% do.

Simple Random Sample

A simple random sample is a subset of individuals chosen from a larger population in such a way that every individual has an equal chance of being selected. This method helps ensure that the sample is representative of the population, reducing bias in the results. In the context of the survey, the 1020 adults surveyed were selected randomly, which supports the validity of the conclusions drawn from the sample.