Discarded Plastic Data Set 42 “Garbage Weight” includes weights (pounds) of discarded plastic from 62 different households. Those 62 weights have a mean of 1.911 pounds and a standard deviation of 1.065 pounds. We want to use a 0.05 level of significance to test the claim that this sample is from a population with a mean less than 2.000 pounds. Identify the null hypothesis and alternative hypothesis.

Verified step by step guidance

1. Step 1: Understand the problem. We are testing a claim about the population mean (μ) of discarded plastic weights. Specifically, we want to test if the population mean is less than 2.000 pounds, using a sample mean of 1.911 pounds, a standard deviation of 1.065 pounds, and a significance level of 0.05.
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 we are testing. In this case: H₀: μ ≥ 2.000 (the population mean is greater than or equal to 2.000 pounds), and Hₐ: μ < 2.000 (the population mean is less than 2.000 pounds).
3. Step 3: Identify the type of test. Since we are testing whether the population mean is less than a specific value, this is a one-tailed test. Additionally, because the population standard deviation is not provided and we are using the sample standard deviation, we will use a t-test.
4. Step 4: Calculate the test statistic. The formula for the t-test statistic is: $t=\frac{\mathrm{x̄}-\mu }{}$₀sn, 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.
5. Step 5: Compare the test statistic to the critical value or use the p-value approach. Determine the critical t-value for a one-tailed test at a 0.05 significance level with degrees of freedom $\mathrm{df}=n-1$. If the test statistic is less than the critical value, or if the p-value is less than 0.05, reject the null hypothesis. Otherwise, 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 population mean of discarded plastic weights is equal to or greater than 2.000 pounds. It is the hypothesis that researchers aim to test against, and it is typically assumed true until evidence suggests otherwise.

Alternative Hypothesis

The alternative hypothesis (H1) represents the statement that contradicts the null hypothesis, suggesting that there is an effect or a difference. In this scenario, it claims that the population mean of discarded plastic weights is less than 2.000 pounds. This hypothesis is what researchers hope to support through statistical testing, indicating a significant finding if the null hypothesis is rejected.

Level of Significance

The level of significance, often denoted as alpha (α), is the threshold for determining whether to reject the null hypothesis. In this case, a significance level of 0.05 means there is a 5% risk of concluding that a difference exists when there is none (Type I error). It helps researchers decide how strong the evidence must be to reject the null hypothesis in favor of the alternative hypothesis.