Question

4 in a marketing study, two different groups of 12 people previewed a new movie. they rated the movie from 10, the best, to 1, the worst. the data is shown below. group a: 8, 7, 1, 6, 8, 5, 5, 8, 8, 1, 7, 8 group b: 8, 7, 1, 6, 5, 5, 7, 2, 8, 1, 7, 6 which statement must be true? circle all that apply. a the mode of group a exceeds the mode of group b by 1. b the mean of group a exceeds the mean of group b by 1. c the median of group a is equal to the median of group b. d the range of group a is equal to the range of group b. the mode of group a exceeds the mode of group b by 1. the mean of group a exceeds the mean of group b by 1. the median of group a is equal to the median of group b. the range of group a is equal to the range of group b.

Explanation:

Step 1: Analyze Mode for Group A and Group B

• Group A data: \(8, 7, 1, 6, 8, 5, 5, 8, 8, 1, 7, 8\)

Count frequencies: \(1\) appears \(2\) times, \(5\) appears \(2\) times, \(6\) appears \(1\) time, \(7\) appears \(2\) times, \(8\) appears \(5\) times.
Mode of Group A: \(8\) (most frequent).

• Group B data: \(8, 7, 1, 6, 5, 5, 7, 2, 8, 1, 7, 6\)

Count frequencies: \(1\) appears \(2\) times, \(2\) appears \(1\) time, \(5\) appears \(2\) times, \(6\) appears \(2\) times, \(7\) appears \(3\) times, \(8\) appears \(2\) times.
Mode of Group B: \(7\) (most frequent).

Difference: \(8 - 7 = 1\). So Statement A is true.

Step 2: Analyze Mean for Group A and Group B

• Group A sum: \(8 + 7 + 1 + 6 + 8 + 5 + 5 + 8 + 8 + 1 + 7 + 8 = 72\)

Mean of Group A: \(\frac{72}{12} = 6\).

• Group B sum: \(8 + 7 + 1 + 6 + 5 + 5 + 7 + 2 + 8 + 1 + 7 + 6 = 63\)

Mean of Group B: \(\frac{63}{12} = 5.25\).

Difference: \(6 - 5.25 = 0.75
eq 1\). So Statement B is false.

Step 3: Analyze Median for Group A and Group B

• Group A (sorted): \(1, 1, 5, 5, 6, 7, 7, 8, 8, 8, 8, 8\)

Median (middle of 12 values, average of 6th and 7th): \(\frac{7 + 7}{2} = 7\).

• Group B (sorted): \(1, 1, 2, 5, 5, 6, 6, 7, 7, 7, 8, 8\)

Median (average of 6th and 7th): \(\frac{6 + 6}{2} = 6\).

\(7
eq 6\), so Statement C is false.

Step 4: Analyze Range for Group A and Group B

• Group A range: Max - Min = \(8 - 1 = 7\).
• Group B range: Max - Min = \(8 - 1 = 7\).

Ranges are equal. So Statement D is true.

Answer:

A. The mode of Group A exceeds the mode of Group B by 1.
D. The range of Group A is equal to the range of Group B.