Answered

A survey was conducted with high school students in each grade to see how many prefer math or English. Some of the data are shown below.




9101112Total

Math		1890

English		40153295

Total		63262967185






Which statements are true? Check all that apply.

  • Twenty-three ninth graders and fifteen eleventh graders prefer math.
  • Fourteen eleventh graders prefer math and eight tenth graders prefer English.
  • Thirty-five twelfth graders prefer math and nine tenth graders prefer English.
  • Twenty-three ninth graders and thirty-five twelfth graders prefer math.
  • Eighteen tenth graders prefer English and fourteen eleventh graders prefer math.

Verified Answer :

  • Calculate missing values in the contingency table using column constraints: \(M_9=23\), \(E_{10}=8\), \(M_{11}=14\), \(M_{12}=35\).
  • Verify row totals to ensure accuracy (\(23+18+14+35=90\) for Math; \(40+8+15+32=95\) for English).
  • Evaluate each statement based on the calculated values.
  • The true statements are:
  • \(\boxed{\text{Fourteen eleventh graders prefer math and eight tenth graders prefer English.}}\)
  • \(\boxed{\text{Twenty-three ninth graders and thirty-five twelfth graders prefer math.}}\)

Explanation

  1. Completing the Contingency Table

To determine the correct statements, we first fill in the missing entries of the provided contingency table. Let \(M_j\) and \(E_j\) be the number of students who prefer Math and English, respectively, in grade \(j\), for \(j \in \{9, 10, 11, 12\}\). The total number of students in each grade corresponds to the column totals:

  • For grade 9: \(M_9 + E_9 = 63\). Given \(E_9 = 40\), we find \(M_9 = 63 - 40 = 23\).
  • For grade 10: \(M_{10} + E_{10} = 26\). Given \(M_{10} = 18\), we find \(E_{10} = 26 - 18 = 8\).
  • For grade 11: \(M_{11} + E_{11} = 29\). Given \(E_{11} = 15\), we find \(M_{11} = 29 - 15 = 14\).
  • For grade 12: \(M_{12} + E_{12} = 67\). Given \(E_{12} = 32\), we find \(M_{12} = 67 - 32 = 35\).
  1. Verifying the Data Consistency

We verify our results by checking the row totals for Math and English:

  • Math row total: \(M_9 + M_{10} + M_{11} + M_{12} = 23 + 18 + 14 + 35 = 90\). This matches the provided total of \(90\).
  • English row total: \(E_9 + E_{10} + E_{11} + E_{12} = 40 + 8 + 15 + 32 = 95\). This matches the provided total of \(95\).

The table values are consistent with the data provided.

  1. Evaluating the Statements

Now we evaluate the given statements using our calculated values:

  • "Twenty-three ninth graders and fifteen eleventh graders prefer math": \(M_9 = 23\) (True), but \(M_{11} = 14\) (False). Therefore, the statement is false.
  • "Fourteen eleventh graders prefer math and eight tenth graders prefer English": \(M_{11} = 14\) (True), and \(E_{10} = 8\) (True). The statement is true.
  • "Thirty-five twelfth graders prefer math and nine tenth graders prefer English": \(M_{12} = 35\) (True), but \(E_{10} = 8\) (False). The statement is false.
  • "Twenty-three ninth graders and thirty-five twelfth graders prefer math": \(M_9 = 23\) (True), and \(M_{12} = 35\) (True). The statement is true.
  • "Eighteen tenth graders prefer English and fourteen eleventh graders prefer math": \(E_{10} = 8\) (False). The statement is false.

The true statements are "Fourteen eleventh graders prefer math and eight tenth graders prefer English" and "Twenty-three ninth graders and thirty-five twelfth graders prefer math".

Examples

Contingency tables are essential in statistics for analyzing relationships between categorical variables. For instance, a school board might use such a table to allocate resources efficiently, such as deciding whether to hire more math tutors for specific grades based on students' preferences. By understanding how to complete these tables, one can effectively organize and interpret data, which is a fundamental skill in fields ranging from public health to market research.

Related Questions