Randomization: Testing a Claim About a Mean
In Exercises 9–12, use the randomization procedure for the indicated exercise.
Section 8-3, Exercise 21 “Lead in Medicine”

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 lead content in the medicine is equal to a specified value (e.g., μ = μ₀), while the alternative hypothesis states that the mean lead content is different (e.g., μ ≠ μ₀).
2. Determine the test statistic to be used. Since this is a test about a mean, the test statistic is usually calculated as: $\frac{\mathrm{x̄}-\mu }{}$₀ s 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. To do this, repeatedly shuffle or resample the data under the assumption that the null hypothesis is true. For each resample, calculate the test statistic (e.g., the mean). This will create a distribution of test statistics under the null hypothesis.
4. Compare the observed test statistic to the randomization distribution. Determine the proportion of simulated test statistics that are as extreme as or more extreme than the observed test statistic. This proportion represents the p-value.
5. Make a decision based on the p-value. If the p-value is less than the significance level (e.g., α = 0.05), reject the null hypothesis. Otherwise, fail to reject the null hypothesis. Interpret the result in the context of the problem, explaining whether there is evidence to support the claim about the mean lead content in the medicine.

Key Concepts

Randomization

Randomization is a statistical technique used to assign subjects to different groups in a way that eliminates bias. It 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 effects of treatments or interventions. This method is crucial in hypothesis testing, particularly when assessing claims about population means.

Hypothesis Testing

Hypothesis testing is a statistical method used to determine whether there is enough evidence to reject a null hypothesis in favor of an alternative hypothesis. It involves calculating a test statistic from sample data and comparing it to a critical value or using a p-value to assess the strength of the evidence. This process is essential for making informed decisions based on sample data regarding population parameters, such as means.

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. In the context of hypothesis testing, the mean is used to represent the expected value of a population, and testing claims about the mean involves comparing sample means to determine if observed differences are statistically significant.