In each exercise,
c. find the test statistic,

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Question

In each exercise,

c. find the test statistic,

In Exercises 1 and 2, use the table, which lists the distribution of educational achievement for people in the United States ages 25 and older. It also lists the results of a random survey for two additional age groups. (Adapted from U.S. Census Bureau)

Table comparing educational attainment across age groups: 25 and older, 30-34, and 65-69 years.

Use the data for 30- to 34-year-olds and 65- to 69-year-olds to test whether age and educational attainment are related. Use α=0.01.

Accepted Answer

Hello, everyone. Let's take a look at this question together. A manager at a call center believes that the type of inquiry, technical, billing, or general, is related to the shift during which the call was made, such as morning, afternoon, or night. A random sample of 90 calls produced the following table, where we have the morning, afternoon, and night shifts, as well as technical billing and general type of inquiry, as well as the totals for both. And we need to calculate the chi square test statistic to test whether the type of inquiry is independent of the shift. So in order to solve this question, we have to recall how to calculate a chi square test statistic in order to test whether the type of inquiry, which is technical, billing, or general, is independent of the shift, which is morning, afternoon, or night. And we are utilizing a random sample of 90 calls which produced the given data table. And so the first step in calculating the chi square test statistic is to state the hypotheses, which includes our null hypothesis that the type of inquiry and shift are independent. And the alternative hypothesis, which is that the type of inquiry and shift are dependent, then the next step is to calculate the expected frequencies, which under independence, the expected count for each cell is given. E is equal to the row total multiplied by the column total divided by the grand total, which using the information provided in the data table, our expected count is equal to 30, multiplied by 30, divided by 90. Which is equal to 10 for all cells, and then we compute the chi square statistic. Using the formula chi square is equal to the sum of the square difference of the observed and expected values divided by the expected values which using the data table that we are provided, we can calculate the square difference of observed and expected values divided by expected values. And then we calculate sum by adding all of the values together, which gives us 2.40 as our chi square statistic, and then we can determine the critical value and the conclusion, which first, the degrees of freedom is calculated as 3 minus 1 in parentheses multiplied by 3 minus 1, which is equal to 4. And using the chi square distribution table, the critical chi square or an alpha equals 0.05, and degrees of freedom equals 4, we get 9.488. So our critical chi square value is 9.488, and using that value, we can make our decision, which since our calculated chi square statistic is equal to 2.4, which is less than 9.488. We know that we failed to reject the null hypothesis. Therefore, our chi square test statistic is equal to 2.4, which tells us that we fail to reject the null hypothesis, which is the type of inquiry and shift, are independent. And looking at our answer choices, we can identify that answer choice A is the correct answer. And all other answer choices are incorrect. I hope you found this video to be helpful. Thank you and goodbye.

Key Concepts

Test Statistic

Hypothesis Testing

Chi-Square Test of Independence

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