The same poll as in Exercise 9 also asked the questions "Did...

The same poll as in Exercise 9 also asked the questions "Did you use the Internet yesterday?" and "What is your educational level?" Is the response to the question about the internet independent of educational level? The data follow: (FIGURE CAN'T COPY) a) Under the null hypothesis, what are the expected values? b) Compute the $\chi^2$ statistic. c) How many degrees of freedom does it have? d) Find the critical value for $\alpha=0.01$. e) What do you conclude?

The same poll as in Exercise 9 also asked the questions "Did you use the Internet yesterday?" and "What is your educational level?" Is the response to the question about the internet independent of educational level? The data follow: (FIGURE CAN'T COPY)
a) Under the null hypothesis, what are the expected values?
b) Compute the $\chi^2$ statistic.
c) How many degrees of freedom does it have?
d) Find the critical value for $\alpha=0.01$.
e) What do you conclude?

Key Concepts

Hypothesis Testing for Independence

This concept involves assessing whether two categorical variables are independent of each other in a population. It typically starts with a null hypothesis stating that the variables are independent and an alternative hypothesis stating that there is an association or dependence between them, with the decision being made based on the observed data and statistical analysis such as the chi-square test.

Expected Values in Contingency Tables

Expected values in a contingency table represent the frequencies that would be expected if the null hypothesis of independence were true. They are calculated using the formula: (row total × column total) / overall total. This concept is crucial because comparing these expected frequencies to the observed counts allows one to assess whether any significant deviation exists.

Chi-Square Test Statistic

The chi-square test statistic is used to measure the discrepancy between the observed counts and the expected counts under the assumption of independence. It is computed using the formula: sum of [(observed - expected)² / expected] over all cells in the table. This statistic, along with the degrees of freedom, helps determine the significance of any deviation from expected values.

Degrees of Freedom

Degrees of freedom in the context of a chi-square test for a contingency table are calculated as (number of rows - 1) multiplied by (number of columns - 1). This number indicates the number of independent comparisons that can be made in the data and is used to reference the chi-square distribution to obtain critical values for hypothesis testing.

Critical Value and Significance Level

The critical value is a threshold determined from the chi-square distribution that corresponds to a chosen significance level (alpha, often set at 0.01, 0.05, etc.). If the computed chi-square statistic exceeds this critical value, the null hypothesis is rejected, indicating that the observed frequencies significantly differ from those expected under the assumption of independence.

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