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Chapter 10: Problem 35

A vaccine is available to prevent the contraction of human papillomavirus (HPV). The Centers for Disease Control and Prevention recommends this vaccination for all young girls in two doses. In a 2015 study reported in the Journal of American College Health, Lee et al. studied vaccination rates among Asian American and Pacific Islander (AAPI) women and non-Latina white women. Data are shown in the table. Test the hypothesis that vaccination rates and race are associated. Use a \(0.05\) significance level. $$\begin{array}{|lcc|}\hline \text { Completed HPV vaccinations } & \text { AAPI } & \text { White } \\\\\hline \text { Yes } & 136 & 1170 \\ \hline \text { No } & 216 & 759 \\\\\hline\end{array}$$

Short Answer

Expert verified
Given the observed counts and expected counts, the chi-square test statistic can be calculated. If the p-value obtained from this test statistic is lower than the significance level (0.05), the null hypothesis that vaccination rates and races are independent is rejected. In that case, it can be concluded that there is a statistical association between vaccination rates and races.

Step by step solution

01

Set up the hypotheses

The null hypothesis (\(H_0\)) is that vaccination rates and race are independent. The alternative hypothesis (\(H_a\)) is that vaccination rates and race are associated.
02

Calculate the expected counts

Expected count for each cell in the table is calculated using the formula: \(E_{i,j} = (n_{i} * m_{j}) / N\), where \(n_i\) is the total of row i, \(m_j\) is the total of column j, and N is the grand total. For example, using this formula, expected count for AAPI women who completed the vaccination (cell 1,1) is calculated as: \((136 + 216) * (136 + 1170) / (136 + 1170 + 216 + 759)\). Perform this calculation for each cell in the table.
03

Calculate the chi-square test statistic

The chi-square test statistic is calculated using the formula: \(X^2 = \sum \frac{(O_{i,j} - E_{i,j})^2}{E_{i,j}}\), where \(O_{i,j}\) is the observed count and \(E_{i,j}\) is the expected count for cell i,j. Calculate this sum for each cell and add them up.
04

Get the chi-square critical value and p-value

The chi-square distribution table is used to find the critical chi-square value for a 0.05 significance level with 1 degree of freedom (since there are 2 rows and 2 columns). The p-value is then obtained from the chi-square distribution with the calculated test statistic. If this p-value is less than the significance level, reject the null hypothesis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Chi-Square Test
The Chi-Square Test is a statistical method used to determine if there is a significant association between two categorical variables. It compares observed counts with expected counts to see if any difference is due to random chance or if it suggests a real association. In our exercise, the goal is to determine if vaccination rates differ by race, using the data from AAPI and white women.

To perform a Chi-Square Test:
  • Start by setting up your hypotheses.
  • Calculate the expected counts for each cell.
  • Compute the Chi-Square statistic for each cell.
  • Sum these values to get the test statistic.
The result tells you if the differences between your observed and expected counts are statistically significant.
Significance Level
The significance level, often denoted as \( \alpha \), represents the probability of rejecting the null hypothesis when it is actually true. This threshold helps determine the risk level of making a Type I error (a false positive). In simple terms, it's the cut-off point where you decide if your test results are indeed significant.

In this exercise, a significance level of 0.05 means there is a 5% risk of concluding that a difference exists when there is no actual association. It provides a benchmark for deciding if the observed association between race and vaccination rates is likely due to more than just chance.
Null Hypothesis
A null hypothesis (\( H_0 \)) is a statement that assumes no effect or association exists between variables in the population. It's the default or "no change" position you begin with in hypothesis testing. In the context of our exercise, the null hypothesis is that the HPV vaccination rates are independent of race, meaning they do not vary between AAPI and white women.

The purpose of the null hypothesis is to provide a basis for comparison. By testing it, we can explore whether there is enough statistical evidence to suggest a real difference or association in the data. If the analysis leads to rejecting the null hypothesis, it implies that there is a significant association worth further investigation.
Expected Counts
Expected counts refer to the frequency of data points we would predict to see in each category of a contingency table if there is no association between the variables. These are calculated based on the total row and column counts and the overall total count.

To find expected counts:
  • Multiply the sum of the row by the sum of the column for each cell.
  • Divide that number by the grand total of all observations.
In the exercise, expected counts help identify whether any observed differences in vaccine rates by race are significant. Comparing these expected expectations with what was actually observed allows us to compute the chi-square statistic and decide if the association is meaningful.

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In a 2015 study by Nanney et al. and published in the Journal of American College Health, a random sample of community college students was asked whether they ate breakfast 3 or more times weekly. The data are reported by gender in the table. $$\begin{array}{lcc}\text { Eat breakfast at least } 3 \times \text { weekly } & \text { Females } & \text { Males } \\\\\hline \text { Yes } & 206 & 94 \\\\\text { No } & 92 & 49 \\\\\hline\end{array}$$ a. Find the row, column, and grand totals, and prepare a table showing these values as well as the counts given. b. Find the percentage of students overall who eat breakfast at least three times weekly. Round off to one decimal place. c. Find the expected number who eat breakfast at least three times weekly for each gender. Round to two decimal places as needed. d. Find the expected number who did not eat breakfast at least three times weekly for each gender. Round to two decimal places as needed. e. Calculate the observed value of the chi-square statistic.
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