Question

multiple choice which value would best fit in the missing cell to suggest there is no evidence of an association between the variables? a 18 b 27 c 54 d 9 digital watch displays date 54 no date display 18 analog watch displays date 27 no date display (blank)

Explanation:

Step1: Analyze the ratio for Digital watches

For Digital watches, the ratio of "Displays date" to "No date display" is \( \frac{54}{18} = 3 \).

Step2: Apply the same ratio to Analog watches

To have no association, the ratio for Analog watches should also be 3. Let the missing value be \( x \). So \( \frac{27}{x} = 3 \), solving for \( x \) gives \( x = \frac{27}{3}=9 \)? Wait, no, wait. Wait, actually, to have no association, the conditional distribution should be the same. The proportion of Digital watches with date display is \( \frac{54}{54 + 18}=\frac{54}{72}=\frac{3}{4} \), and without is \( \frac{18}{72}=\frac{1}{4} \). For Analog, if we want the same proportion, the number with no date display should be such that \( \frac{x}{27 + x}=\frac{1}{4} \)? Wait, no, maybe simpler: the ratio of date display to no date for Digital is 54:18 = 3:1. So for Analog, date display is 27, so no date should be 9? Wait, but let's check the options. Wait, maybe the expected ratio is that the ratio of Digital's date to Analog's date is equal to Digital's no date to Analog's no date. So \( \frac{54}{27}=\frac{18}{x} \), so \( 2=\frac{18}{x} \), so \( x = 9 \)? Wait, but the options are 18,27,54,9. Wait, no, maybe I made a mistake. Wait, the problem says "no evidence of an association", which means the two variables (watch type and date display) are independent. So the expected frequency is calculated as (row total column total)/grand total. Let's calculate grand total: Digital date:54, Digital no date:18, Analog date:27, Analog no date:x. Grand total = 54 + 18 + 27 + x = 99 + x. Row total for Digital: 54 + 18 = 72. Row total for Analog:27 + x. Column total for date:54 + 27 = 81. Column total for no date:18 + x. For independence, (Digital date) = (Digital row total date column total)/grand total. So 54 = (72 * 81)/(99 + x). But that's complicated. Alternatively, the ratio of date to no date for Digital is 54:18 = 3:1. So for Analog, date is 27, so no date should be 9 (since 27:9 = 3:1), so that the ratio is the same, meaning no association. So the missing value is 9.

Answer:

D. 9