students in a fitness class each completed a one-mile walk or run. the list shows the time it took each person to complete the mile. each time is rounded to the nearest half-minute.
5.5, 6, 7, 10, 7.5, 8, 9.5, 9, 8.5, 8, 7, 7.5, 6, 6.5, 5.5
which statements are true about a histogram with one-minute increments representing the data? select three options.
☐ a histogram will show that the mean time is approximately equal to the median time of 7.5 minutes.
☐ the histogram will have a shape that is left skewed.
☐ the histogram will show that the mean time is greater than the median time of 7.4 minutes.
☐ the shape of the histogram can be approximated with a normal curve.
☐ the histogram will show that most of the data is centered between 6 minutes and 9 minutes.

Question

Accepted Answer

To solve this, we analyze the data and histogram properties:

### Step 1: Organize the Data
First, list the times: $ 5.5, 6, 7, 10, 7.5, 8, 9.5, 9, 8.5, 8, 7, 7.5, 6, 6.5, 5.5 $.
Sort them: $ 5.5, 5.5, 6, 6, 6.5, 7, 7, 7.5, 7.5, 8, 8, 8.5, 9, 9.5, 10 $.

### Step 2: Analyze Each Statement

1. **"A histogram will show that the mean time is approximately equal to the median time of 7.5 minutes."**
   - Median: Middle value of sorted data (8th term) is $ 7.5 $.
   - Mean: $ \frac{5.5+5.5+6+6+6.5+7+7+7.5+7.5+8+8+8.5+9+9.5+10}{15} $.
     Calculate sum: $ 5.5(2) + 6(2) + 6.5 + 7(2) + 7.5(2) + 8(2) + 8.5 + 9 + 9.5 + 10 = 11 + 12 + 6.5 + 14 + 15 + 16 + 8.5 + 9 + 9.5 + 10 = 111 $.
     Mean: $ \frac{111}{15} = 7.4 $.
     Mean ($ 7.4 $) is close to median ($ 7.5 $)—**True**.

2. **"The histogram will have a shape that is left - skewed."**
   - Left - skewed: Tail on the left (low values). But we have a high outlier (10), and most data is between 5.5 - 9.5. The tail is on the right (high values, like 10), so it is right - skewed—**False**.

3. **"The histogram will show that the mean time is greater than the median time of 7.4 minutes."**
   - Mean is $ 7.4 $, median is $ 7.5 $. So mean ($ 7.4 $) < median ($ 7.5 $)—**False** (Wait, correction: Earlier mean calculation was wrong. Let's recalculate sum:
     $ 5.5+5.5 = 11 $; $ 6 + 6 = 12 $; $ 6.5 $; $ 7+7 = 14 $; $ 7.5+7.5 = 15 $; $ 8+8 = 16 $; $ 8.5 $; $ 9 $; $ 9.5 $; $ 10 $.
     Sum: $ 11+12 = 23 $; $ 23 + 6.5 = 29.5 $; $ 29.5+14 = 43.5 $; $ 43.5+15 = 58.5 $; $ 58.5+16 = 74.5 $; $ 74.5+8.5 = 83 $; $ 83+9 = 92 $; $ 92+9.5 = 101.5 $; $ 101.5+10 = 111.5 $.
     Mean: $ \frac{111.5}{15}\approx7.43 $, median is $ 7.5 $. Still mean < median. So this statement is false. Wait, maybe I misread. Wait the option says "median time of 7.4 minutes"—no, median is 7.5. Maybe a typo, but let's check other options.)

4. **"The shape of the histogram can be approximated with a normal curve."**
   - Normal curve is symmetric. Our data has a right tail (outlier 10), but most data is clustered around the center (5.5 - 9.5), so it can be approximated as normal (since the skew is not extreme)—**True**.

5. **"The histogram will show that most of the data is centered between 6 minutes and 9 minutes."**
   - Count data in 6 - 9: $ 6, 6, 6.5, 7, 7, 7.5, 7.5, 8, 8, 8.5, 9 $ (11 out of 15 values). Most data is here—**True**.

Wait, initial mean calculation error. Let's re - evaluate:

- Correct sum: $ 5.5+5.5 = 11 $; $ 6+6 = 12 $; $ 6.5 $; $ 7+7 = 14 $; $ 7.5+7.5 = 15 $; $ 8+8 = 16 $; $ 8.5 $; $ 9 $; $ 9.5 $; $ 10 $.
  Sum: $ 11 + 12=23 $; $ 23+6.5 = 29.5 $; $ 29.5+14 = 43.5 $; $ 43.5+15 = 58.5 $; $ 58.5+16 = 74.5 $; $ 74.5+8.5 = 83 $; $ 83+9 = 92 $; $ 92+9.5 = 101.5 $; $ 101.5+10 = 111.5 $.
  Mean: $ \frac{111.5}{15}\approx7.43 $, median is $ 7.5 $.

Now, re - check statements:

1. "A histogram will show that the mean time is approximately equal to the median time of 7.5 minutes."
   Mean ($ \approx7.43 $) is close to median ($ 7.5 $)—**True**.

2. "The histogram will have a shape that is left - skewed."
   Data has a right tail (10), so right - skewed—**False**.

3. "The histogram will show that the mean time is greater than the median time of 7.4 minutes."
   Mean ($ \approx7.43 $) is slightly greater than 7.4, and median is 7.5. Wait, the option says "median time of 7.4 minutes"—maybe a typo. But mean ($ \approx7.43 $) > 7.4, and median is 7.5. But let's check the other options.

4. "The shape of the histogram can be approximated with a normal curve."
   The data is mostly clustered in the middle with a slight right tail, which can be approximated as normal (since normal distribution allows for slight skewness in real - world data)—**True**.

5. "The histogram will show that most of the data is centered between 6 minutes and 9 minutes."
   As calculated, 11 out of 15 values are between 6 - 9—**True**.

So the three true statements are:

- A histogram will show that the mean time is approximately equal to the median time of 7.5 minutes.
- The shape of the histogram can be approximated with a normal curve.
- The histogram will show that most of the data is centered between 6 minutes and 9 minutes.

# Answer:
- A histogram will show that the mean time is approximately equal to the median time of 7.5 minutes.
- The shape of the histogram can be approximated with a normal curve.
- The histogram will show that most of the data is centered between 6 minutes and 9 minutes.

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QUESTION IMAGE

Question

Explanation:

Step 1: Organize the Data

Step 2: Analyze Each Statement

Answer: