Randomization: Testing a Claim About a Mean
In Exercises 9–12, use the randomization procedure for the indicated exercise.
Section 8-3, Exercise 23 “Cell Phone Radiation”

Verified step by step guidance

1. Identify the null hypothesis (H₀) and the alternative hypothesis (H₁). For this problem, the null hypothesis typically states that the mean is equal to a specific value (e.g., μ = μ₀), while the alternative hypothesis states that the mean is different (e.g., μ ≠ μ₀).
2. Determine the test statistic to be used. For testing a claim about a mean, the test statistic is often calculated as: $\frac{\mathrm{x̄}-\mu ₀}{\frac{s}{\sqrt{n}}}$, where $\mathrm{x̄}$ is the sample mean, $\mu ₀$ is the hypothesized population mean, $s$ is the sample standard deviation, and $n$ is the sample size.
3. Simulate the randomization distribution. Randomly shuffle or resample the data under the assumption that the null hypothesis is true. This involves creating a large number of simulated datasets where the mean is fixed at the hypothesized value, $\mu ₀$.
4. Calculate the test statistic for each simulated dataset. This will create a distribution of test statistics under the null hypothesis, which is called the randomization distribution.
5. Compare the observed test statistic to the randomization distribution. Determine the p-value by finding the proportion of simulated test statistics that are as extreme or more extreme than the observed test statistic. Use this p-value to decide whether to reject or fail to reject the null hypothesis based on the significance level (e.g., α = 0.05).

Key Concepts

Randomization

Randomization is a statistical technique used to assign subjects to different groups in a way that eliminates bias. This process ensures that each participant has an equal chance of being placed in any group, which helps to create comparable groups and allows for valid inferences about the population. In hypothesis testing, randomization can be used to simulate the distribution of a test statistic under the null hypothesis.

Mean

The mean, often referred to as the average, is a measure of central tendency that summarizes a set of values by dividing the sum of those values by the number of observations. It provides a single value that represents the entire dataset, making it easier to understand the overall trend. In hypothesis testing, the mean is often compared to a hypothesized value to determine if there is enough evidence to reject the null hypothesis.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating a null hypothesis (H0) and an alternative hypothesis (H1), then using sample data to determine whether to reject H0. The process includes calculating a test statistic and comparing it to a critical value or using a p-value to assess the strength of the evidence against H0, guiding conclusions about the population parameter.