Testing Claims About Variation
In Exercises 5–16, test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim. Assume that a simple random sample is selected from a normally distributed population.


Minting of Pennies Data Set 40 “Coin Weights” lists weights (grams) of pennies minted after 1983. Here are the statistics for those weights: n = 37, xbar = 2.49910 g, s = 0.01648 g . Use a 0.05 significance level to test the claim that the sample is from a population of pennies with weights having a standard deviation greater than 0.01000 g.

Verified step by step guidance

1. Step 1: Define the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis is that the population standard deviation is equal to 0.01000 g (H₀: σ = 0.01000 g). The alternative hypothesis is that the population standard deviation is greater than 0.01000 g (H₁: σ > 0.01000 g). This is a one-tailed test.
2. Step 2: Identify the test statistic to use. Since we are testing a claim about the population standard deviation, we use the chi-square test for variance. The test statistic is calculated using the formula: ${\chi }^2=\frac{\left(n-1\right)s^2}{{\sigma }^2},\; where\; n\; is\; the\; sample\; size,\; s\; is\; the\; sample\; standard\; deviation,\; and\; \sigma \; is\; the\; hypothesized\; population\; standard\; deviation.$
3. Step 3: Compute the degrees of freedom (df). The degrees of freedom for the chi-square test is given by df = n - 1. In this case, df = 37 - 1 = 36.
4. Step 4: Determine the critical value or P-value. Using a chi-square distribution table or statistical software, find the critical value for a one-tailed test with df = 36 at a significance level of 0.05. Alternatively, calculate the P-value based on the test statistic.
5. Step 5: Compare the test statistic to the critical value or compare the P-value to the significance level. If the test statistic exceeds the critical value or if the P-value is less than 0.05, reject the null hypothesis. Otherwise, fail to reject the null hypothesis. State the conclusion: If the null hypothesis is rejected, conclude that the population standard deviation is greater than 0.01000 g. If the null hypothesis is not rejected, conclude that there is insufficient evidence to support the claim.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating two competing hypotheses: the null hypothesis (H0), which represents a statement of no effect or no difference, and the alternative hypothesis (H1), which indicates the presence of an effect or difference. The goal is to determine whether there is enough evidence in the sample data to reject the null hypothesis in favor of the alternative.

P-value

The P-value is a measure that helps determine the strength of the evidence against the null hypothesis. It represents the probability of obtaining a test statistic at least as extreme as the one observed, assuming that the null hypothesis is true. A smaller P-value indicates stronger evidence against the null hypothesis, and if it is less than the predetermined significance level (e.g., 0.05), the null hypothesis is rejected.

Standard Deviation

Standard deviation is a statistic that quantifies the amount of variation or dispersion in a set of values. In the context of hypothesis testing, it is crucial for understanding the spread of the data and is used to calculate the test statistic. A larger standard deviation indicates more variability in the data, which can affect the conclusions drawn from hypothesis tests, particularly when assessing claims about population parameters.