Performing a Chi-Square Independence Test In ...

Question

Performing a Chi-Square Independence Test In Exercises 13–28, perform the indicated chi-square independence test by performing the steps below.

a. Identify the claim and state H₀ and Hₐ

b. Determine the degrees of freedom, find the critical value, and identify the rejection region.

c. Find the chi-square test statistic.

d. Decide whether to reject or fail to reject the null hypothesis.

e. Interpret the decision in the context of the original claim.

Alcohol-Related Accidents The contingency table shows the results of a random sample of fatally injured passenger vehicle drivers (with blood alcohol concentrations greater than or equal to 0.08) by age and gender. At α=0.05, can you conclude that age is related to gender in such alcohol-related accidents? (Adapted from Insurance Institute for Highway Safety)

Accepted Answer

All right, hello, everyone. So, this question says, a marketing firm conducted a survey of individuals who use smartphones for more than 4 hours daily. The table below shows the number of such users categorized by age group and type of primary usage. At alpha equals 0.05, can you conclude that the type of smartphone usage is related to age group? Perform a chi square test of independence at alpha equals 0.05 to determine whether primary smartphone usage is dependent on age group. And here we have 4 different answer choices labeled A through D. All right, so first, let's identify our hypotheses. Now, the null hypothesis, Hno would state that there is no correlation between age group and smartphone usage. Which would mean that these two factors are independent. The alternative hypothesis, on the other hand, would state instead that there is a correlation. Meaning that the age group and type of smartphone usage are not independent. So for the rest of this problem, we are going to have to look at the. Data table that we are provided here. Because for the chi square test of independence, recall that we have to compare between. The observed frequencies listed here in the table, and the expected frequencies. So here The expected frequency E. Is equal to the row total. Added to the column total. Divided by The grand total. So repeating this calculation for each of. The data points that were given will give you an expected frequency to accompany the observed frequency that we see in the table itself. So after you obtain the expected frequency, you can now proceed with the test statistic. Because the test statistic for the chi square test is equal to the sum. Of all squared differences between observed and expected frequencies divided by the expected frequency. So just to clarify this calculation, for each data point, You take the observed frequency and subtract the expected frequency for that data point. You then square the difference and then divide the squared difference by the expected frequency. So you repeat this for. For all of your data points and then find the sum of all of those values. To illustrate this point. Let's compute one cell. So let's say for example, That we pick the cell in the 18 to 24 age group that corresponds to productivity. Oh, the expected frequency is equal to 12, and E, according to our earlier calculations, is equal to 27.33. So what you do you subtract 27.33 from 12. So that's 12 subtracted by 27.33. Squared And then divided by 27.33. This gives you approximately 8.60. And so you repeat this calculation for all of the cells in our data table. And ultimately the sum of all of them which corresponds to the chi squired statistic. Is equal to 40.4124. Which now brings you to your critical value. Now, because we're given a contingency table, notice that we have 4 different age groups and 4 different categories of smartphone usage. So the degrees of freedom is equal to 4 subtracted by 1 multiplied by 4 subtracted by 1, which gives you 9. So now Our chi square critical value at alpha equals 0.05 and 9 degrees of freedom. is equal to 16.919. Ultimately, we can see that our test statistic is greater than our critical value. Which means that we can reject. The null hypothesis. So now, what conclusion can you now make? First, let me go back to our answer choices. The correct answer is going to be option B, which reads as follows. Reject the null hypothesis. There is a statistically significant relationship between age group and smartphone usage type. And there you have it. So with that being said, thank you so very much for watching, and I hope you found this helpful.

Key Concepts

Chi-Square Independence Test

Null and Alternative Hypotheses

Degrees of Freedom and Critical Value

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