Objectives

Previously…

Confidence Interval for One Proportion

\[\hat{p} \pm z^{\star} \text{SE}_{\hat{p}}\]

\[ \begin{aligned} \hat{p} & \longrightarrow \text{sample proportion (or the point estimate)} \\ z^{\star} & \longrightarrow \text{critical z-score at a given confidence level} \\ \text{SE}_{\hat{p}} & \longrightarrow \text{standard error of the sampling distribution} \\ \end{aligned} \]

Hypothesis Testing for One Proportion

\[ \begin{aligned} p & \longrightarrow \text{population proportion} \\ \hat{p} & \longrightarrow \text{sample proportion (or the point estimate)} \\ H_0: p = p_0 & \longrightarrow \text{null hypothesis} \\ H_A: p \ne p_0 & \longrightarrow \text{alternative hypothesis (can be } < \text{ or } > \text{)} \\ z & \longrightarrow \text{test statistic} \\ \text{SE}_{p} & \longrightarrow \text{standard error of the null distribution} \\ \end{aligned} \]

The Chi-Squared Statistic

The Chi-Square statistic is used in hypothesis testing to determine whether observed data differs significantly from expected data. Commonly used for categorical data analysis.

\[\chi^2 = \sum \frac{(O-E)^2}{E}\]

• \(O\) is the observed frequency
• \(E\) is the expected frequency

The Chi-Squared Distribution

Example 1

Consider the following problem description.

Students in grades 4-6 were asked whether good grades, athletic ability, or popularity was most important to them. A two-way table separating the students by grade and by choice of most important factor is shown below. Do these data provide evidence to suggest that goals vary by grade?

Source: Popular Kids Dataset. This is from a 1992 study and was revisited 30 years later.

Example 1: The Chi-Squared Test for Independence (1/2)

• The null and alternative Hypothesis \[H_0: \text{Grade and goals are independent. Goals do not vary by grade.}\] \[H_A: \text{Grade and goals are dependent. Goals vary by grade}\]

Example 1: The Chi-Squared Test for Independence (2/2)

• The Chi-Squared test statistic \[\chi^2_{k} = \sum_{i=1}^n \frac{(O_i - E_i)^2}{E_i}\]
  • \(O_i\) is the number of observations of type \(i\)
  • \(E_i\) is the expected frequency of type \(i\)
  • \(n\) is the number of cells in the table
  • \(k = (R-1)(C-1)\) is the degrees of freedom where \(R\) is the number of rows and C is the number of columns.

Example 1: Computing the \(\chi^2\) statistic - Expected Frequency (1/3)

• Start with the expected frequency of type \(i\)

Example 1: Computing the \(\chi^2\) statistic - Expected Frequency (2/3)

• Question - What is the expected count for the highlighted cell?

• \[\color{green}{E_{5th,Popular} = \frac{(176)(141)}{478} = 52}\]

Example 1: Computing the \(\chi^2\) statistic - Expected Frequency (3/3)

• The expected frequency for each \(\color{blue}{[cell]}\).

Example 1: Computing the \(\chi^2\) statistic

• The \(\chi^2\) statistic. \[\chi^2_{k} = \frac{(63-61)^2}{61} + \frac{(31-35)^2}{35} + \cdots + \frac{(32-34)^2}{34} = 0.967\]

• Degrees of freedom. \[k = (3-1) \times (3-1) = 2(2) = 4\]

Example 1: Computing the p-value

• \(\chi^2_{k} = 1.3121\) and \(k = 4\)

• We can use the pchisq function in R.

df <- 4
1-pchisq(0.967,df)

## [1] 0.9147579

The p-value of 0.9148.

• Note that in a chi-squared analysis, the p-value is the probability of obtaining a chi-square as large or larger than that in the current experiment.

Example 1: Conclusion

• Do these data provide evidence to suggest that goals vary by grade? \[H_0: \text{Grade and goals are independent. Goals do not vary by grade.}\] \[H_A: \text{Grade and goals are dependent. Goals vary by grade}\]

• Since the p-value is large, we fail to reject \(H_0\). The data do not provide convincing evidence that grade and goals are dependent. It doesn’t appear that goals vary by grade.

Example 2

The problem shown below was taken and slightly modified from your textbook OpenIntro: Introduction to Modern Statistics Section 18.4. Consider the research study described below.

Coffee and Depression.

Researchers conducted a study investigating the relationship between caffeinated coffee consumption and risk of depression in women. They collected data on 50,739 women free of depression symptoms at the start of the study in the year 1996, and these women were followed through 2006. The researchers used questionnaires to collect data on caffeinated coffee consumption, asked each individual about physician- diagnosed depression, and also asked about the use of antidepressants. The table below shows the distribution of incidences of depression by amount of caffeinated coffee consumption. Lucas et al. 2011

1. Compute the test statistic. What is the p-value?

2. What is the conclusion of the hypothesis test?

Example 2: Results

• \(\chi^2 =20.932\) and degrees of freedom is \(k =4\)

• p-value:

## [1] 0.0003266524

• Therefore, p-value is small and we reject \(H_0\). The data provide convincing evidence to suggest that caffeinated coffee consumption and depression in women are associated.

Activity: Test Independence for Two-Way Tables

1. Make sure you have a copy of the F 3/21 Worksheet. This will be handed out physically and it is also digitally available on Moodle.
2. Work on your worksheet by yourself for 10 minutes. Please read the instructions carefully. Ask questions if anything need clarifications.
3. Get together with another student.
4. Discuss your results.
5. Submit your worksheet on Moodle as a .pdf file.

References