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.


Birth Weights A simple random sample of birth weights of 30 girls has a standard deviation of 829.5 g. Use a 0.01 significance level to test the claim that birth weights of girls have the same standard deviation as birth weights of boys, which is 660.2 g (based on Data Set 6 “Births” in Appendix B).

Verified step by step guidance

1. Step 1: Define the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis states that the standard deviation of birth weights for girls is equal to that of boys (σ = 660.2 g). The alternative hypothesis states that the standard deviation of birth weights for girls is not equal to that of boys (σ ≠ 660.2 g).
2. Step 2: Identify the test statistic to use. Since this is a test about the population standard deviation, use the chi-square test for variance. The test statistic is given by the formula: χ² = ((n - 1) * s²) / σ₀², where n is the sample size, s is the sample standard deviation, and σ₀ is the hypothesized population standard deviation.
3. Step 3: Calculate the degrees of freedom (df). The degrees of freedom for the chi-square test is given by df = n - 1, where n is the sample size. In this case, n = 30, so df = 30 - 1 = 29.
4. Step 4: Determine the critical value(s) or P-value. Since this is a two-tailed test (σ ≠ 660.2 g), find the critical chi-square values for a significance level of 0.01 and df = 29. Use a chi-square distribution table or statistical software to find these values. Alternatively, calculate the P-value based on the test statistic.
5. Step 5: Compare the test statistic to the critical value(s) or compare the P-value to the significance level (α = 0.01). If the test statistic falls outside the range of the critical values or if the P-value is less than 0.01, reject the null hypothesis. Otherwise, fail to reject the null hypothesis. State the conclusion in the context of the problem: whether there is sufficient evidence to support the claim that the standard deviation of birth weights for girls is different from that of boys.

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., 0.01), the null hypothesis is rejected.

Standard Deviation and Variance

Standard deviation is a statistic that quantifies the amount of variation or dispersion in a set of data values. It is the square root of the variance, which measures the average squared deviation of each data point from the mean. In hypothesis testing, comparing the standard deviations of two groups (e.g., birth weights of girls and boys) can help assess whether there is a significant difference in variability between the two populations.