In Exercises 5–20, conduct the hypothesis test and provide the test statistic and the P-value and/or critical value, and state the conclusion.


Bias in Clinical Trials? Researchers investigated the issue of race and equality of access to clinical trials. The following table shows the population distribution and the numbers of participants in clinical trials involving lung cancer (based on data from “Participation in Cancer Clinical Trials,” by Murthy, Krumholz, and Gross, Journal of the American Medical Association, Vol. 291, No. 22). Use a 0.01 significance level to test the claim that the distribution of clinical trial participants fits well with the population distribution. Is there a race/ethnic group that appears to be very underrepresented?

Verified step by step guidance

1. Step 1: Define the null and alternative hypotheses. The null hypothesis (H₀) states that the distribution of clinical trial participants matches the population distribution. The alternative hypothesis (H₁) states that the distribution of clinical trial participants does not match the population distribution.
2. Step 2: Calculate the expected frequencies for each race/ethnic group based on the population distribution. Multiply the total number of participants in the clinical trials (sum of all observed frequencies) by the percentage distribution for each group. For example, for White non-Hispanic: Expected frequency = Total participants × 75.6%.
3. Step 3: Compute the test statistic using the chi-square formula: χ² = Σ((Observed - Expected)² / Expected). For each race/ethnic group, subtract the expected frequency from the observed frequency, square the result, divide by the expected frequency, and sum these values across all groups.
4. Step 4: Determine the critical value or P-value. Use the chi-square distribution table with degrees of freedom (df = number of categories - 1) and the significance level (α = 0.01) to find the critical value. Alternatively, calculate the P-value using statistical software or a calculator.
5. Step 5: Compare the test statistic to the critical value or interpret the P-value. If the test statistic exceeds the critical value or if the P-value is less than 0.01, reject the null hypothesis. Otherwise, fail to reject the null hypothesis. Based on the conclusion, identify if any race/ethnic group appears to be underrepresented by comparing observed and expected frequencies.

Key Concepts

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) that represents no effect or no difference, and an alternative hypothesis (H1) that indicates the presence of an effect or difference. The test assesses the evidence against H0 using a test statistic and a significance level, leading to a conclusion about whether to reject or fail to reject H0.

P-value

The P-value is a measure that helps determine the strength of the evidence against the null hypothesis in hypothesis testing. 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 H0, and if it is less than the predetermined significance level (e.g., 0.01), the null hypothesis is rejected.

Chi-Square Test for Goodness of Fit

The Chi-Square Test for Goodness of Fit is a statistical test used to determine if the observed frequency distribution of a categorical variable matches an expected distribution. In this context, it assesses whether the distribution of clinical trial participants aligns with the population distribution by comparing the observed counts in each category to the expected counts based on population proportions. A significant result indicates that the observed distribution significantly differs from what was expected.