Question

1. given the following data set:

65, 70, 70, 70, 70, 80, 80, 80, 85, 90, 90, 95, 95, 95, 100
which representations are correct for this data set?
i: dot plot with 65 (1 dot), 70 (4 dots), 80 (3 dots), 85 (1 dot), 90 (2 dots), 95 (3 dots), 100 (1 dot); ii: box plot with scale 65–100; iii: histogram with intervals 61–71, 71–81, 81–91, 91–101, frequencies: 61–71 (5), 71–81 (3), 81–91 (3), 91–101 (4)

1. i and ii
2. i and iii, only
3. ii and iii, only
4. i, ii, and iii

Explanation:

Step1: Analyze Representation I (Dot Plot)

Count the number of dots for each value: 65 (1), 70 (4), 80 (3), 85 (1), 90 (2), 95 (3), 100 (1). The given data set has 1, 4, 3, 1, 2, 3, 1 occurrences for these values respectively. So Representation I matches the data.

Step2: Analyze Representation II (Box Plot)

First, find the five - number summary. The data set is \(65,70,70,70,70,80,80,80,85,90,90,95,95,95,100\) (n = 15). The median (Q2) is the 8th value, which is 80. The lower half: \(65,70,70,70,70,80,80\) (n = 7), Q1 is the 4th value, 70. The upper half: \(85,90,90,95,95,95,100\) (n = 7), Q3 is the 4th value from the upper half, 95. The minimum is 65, maximum is 100. The box plot should have minimum = 65, Q1 = 70, median = 80, Q3 = 95, maximum = 100. The given box plot (II) seems to have the correct positions for these values (the box is between Q1 and Q3, whiskers from min to max). So Representation II matches.

Step3: Analyze Representation III (Histogram)

The intervals: 61 - 71, 71 - 81, 81 - 91, 91 - 101.

• For 61 - 71: Values are 65, 70,70,70,70. So frequency = 5.
• For 71 - 81: Values are 80,80,80. So frequency = 3.
• For 81 - 91: Values are 85,90,90. So frequency = 3.
• For 91 - 101: Values are 95,95,95,100. So frequency = 4.

The histogram (III) has frequencies 5, 3, 3, 4 for these intervals respectively, which matches the data.

Answer:

1. I, II, and III