Question

1. the data set represents the number of hours a small group of students met to study each day for twelve days: 2, 3, 3, 4, 4, 5, 6, 6, 6, 7, 8, 9. select all the values that could represent the typical number of hours the students met to study each day.

a. 4.5
b. 5
c. 5.25
d. 5.5
e. 6.5

1. the dot plot displays the lengths of pencils (in inches) used by students in a class. what is the median?

dot plot with 5 (1 dot), 5.5 (2 dots), 6 (4 dots), 6.5 (1 dot), 7 (1 dot)

Explanation:

Question 2

Step1: Calculate the mean

First, find the sum of the data set: \(2 + 3 + 3 + 4 + 4 + 5 + 6 + 6 + 6 + 7 + 8 + 9\). Let's compute that: \(2+3=5\), \(5+3=8\), \(8+4=12\), \(12+4=16\), \(16+5=21\), \(21+6=27\), \(27+6=33\), \(33+6=39\), \(39+7=46\), \(46+8=54\), \(54+9=63\). There are 12 data points, so the mean is \(\frac{63}{12} = 5.25\).

Step2: Find the median

Since there are 12 (even) data points, the median is the average of the 6th and 7th values. The data set in order: 2, 3, 3, 4, 4, 5, 6, 6, 6, 7, 8, 9. The 6th value is 5, the 7th is 6. So median is \(\frac{5 + 6}{2} = 5.5\).

Step3: Analyze the mode

The mode is the most frequent value, which is 6 (appears 3 times). But the question is about "typical" values, which can include mean, median, or values close. Let's check the options:

• A. 4.5: Too low, not typical.
• B. 5: Close to median (5.5) and lower half, but let's see. Wait, mean is 5.25, median 5.5, mode 6. Let's re - evaluate. Wait, the data: 2,3,3,4,4,5,6,6,6,7,8,9. The mean is 5.25, median 5.5, mode 6. So 5 (B), 5.25 (C), 5.5 (D) are reasonable. Wait, wait, let's recalculate the mean: \(2 + 3+3 + 4+4 + 5+6+6+6+7+8+9\). Let's add again: 2+3=5, +3=8, +4=12, +4=16, +5=21, +6=27, +6=33, +6=39, +7=46, +8=54, +9=63. 63 divided by 12 is 5.25. Correct. Median: 12 data points, positions 6 and 7: 5 and 6, average is 5.5. So typical values can be mean (5.25), median (5.5), or maybe 5 (since it's a middle value in the lower half? Wait, no, 5 is the 6th value? Wait no, the data is ordered: 1:2, 2:3, 3:3, 4:4, 5:4, 6:5, 7:6, 8:6, 9:6, 10:7, 11:8, 12:9. So 6th is 5, 7th is 6. So median is (5 + 6)/2 = 5.5. So mean is 5.25, median 5.5, mode 6. So the options that make sense: B (5) is close to the lower middle, C (5.25) is mean, D (5.5) is median. Wait, but let's check the options again. The question says "select all the values that could represent the typical number". So:

• A. 4.5: The data has values from 2 - 9, but 4.5 is lower than mean and median, not typical.
• B. 5: The 6th value, a middle value in the ordered data, could be considered typical.
• C. 5.25: The mean, which is a measure of central tendency, so typical.
• D. 5.5: The median, a measure of central tendency, typical.
• E. 6.5: Higher than mean, median, and mode (mode is 6), not typical.

So the correct options are B, C, D.

Step1: Count the number of dots

First, we need to count the number of dots (pencil lengths). Let's assume the dot plot has: at 5: 1 dot, at 5.5: 2 dots, at 6: 4 dots, at 6.5: 1 dot, at 7: 1 dot. Wait, let's count: 1 (at 5) + 2 (at 5.5) + 4 (at 6) + 1 (at 6.5) + 1 (at 7) = 9 dots. Wait, no, maybe I miscounted. Wait, the dot plot: 5 has 1, 5.5 has 2, 6 has 4, 6.5 has 1, 7 has 1. Total: 1+2 + 4+1+1=9. Wait, but 9 is odd, so the median is the (9 + 1)/2=5th value. Let's order the data:

• 5 (1 time)
• 5.5 (2 times: so values 2nd and 3rd)
• 6 (4 times: values 4th, 5th, 6th, 7th)
• 6.5 (1 time: 8th)
• 7 (1 time: 9th)

So the 5th value is 6.

Wait, let's list them in order: 5, 5.5, 5.5, 6, 6, 6, 6, 6.5, 7. Wait, no, 1 dot at 5: [5], 2 dots at 5.5: [5, 5.5, 5.5], 4 dots at 6: [5, 5.5, 5.5, 6, 6, 6, 6], 1 dot at 6.5: [5, 5.5, 5.5, 6, 6, 6, 6, 6.5], 1 dot at 7: [5, 5.5, 5.5, 6, 6, 6, 6, 6.5, 7]. Now, the 5th value (since n = 9, median is (9+1)/2 = 5th term) is 6.

Answer:

B. 5, C. 5.25, D. 5.5

Question 3