Question

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.

Explanation:

First, let's list out the data points: 5.5, 6, 7, 10, 7.5, 8, 9.5, 9, 8.5, 8, 7, 7.5, 6, 6.5, 5.5. Let's 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 1: Analyze the mean and median

- **Median**: There are 15 data points, so the median is the 8th value. The 8th value is 7.5.
- **Mean**: Let's calculate the sum.
- 5.5 + 5.5 = 11; 6 + 6 = 12; 6.5 = 6.5; 7 + 7 = 14; 7.5 + 7.5 = 15; 8 + 8 = 16; 8.5 = 8.5; 9 = 9; 9.5 = 9.5; 10 = 10.
- Sum = 11 + 12 + 6.5 + 14 + 15 + 16 + 8.5 + 9 + 9.5 + 10 = Let's add step by step: 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 = 111.5 / 15 ≈ 7.43, which is approximately 7.4, close to 7.5. So the mean is approximately equal to the median (7.5) since 7.43 is close to 7.5. So the first statement "A histogram will show that the mean time is approximately equal to the median time of 7.5 minutes" is true.

Step 2: Analyze the shape (skewness)

- The data has a few larger values (9, 9.5, 10) but also smaller values (5.5, 5.5). Wait, when we sorted, the lower end: 5.5,5.5,6,6,6.5 (left side), middle:7,7,7.5,7.5, and right side:8,8,8.5,9,9.5,10. Wait, actually, the right tail has a few larger values (9,9.5,10) but the left tail is shorter? Wait, no. Wait, the number of data points: left of median (7.5): 7 values (5.5,5.5,6,6,6.5,7,7), right of median: 7 values (8,8,8.5,9,9.5,10,7.5? Wait no, sorted list: positions 1 - 15: 1:5.5, 2:5.5, 3:6, 4:6, 5:6.5, 6:7, 7:7, 8:7.5 (median), 9:7.5, 10:8, 11:8, 12:8.5, 13:9, 14:9.5, 15:10. So left of median (positions 1 - 7): 7 values, right of median (positions 9 - 15): 7 values (position 8 is median). Wait, the right side has values 7.5,8,8,8.5,9,9.5,10. The left side has 5.5,5.5,6,6,6.5,7,7. So the right tail is longer (has 10,9.5,9 which are further from the median than the left tail's 5.5,5.5,6). Wait, no, left tail: the smallest values are 5.5 (two times), 6 (two times), 6.5. Right tail: 8.5,9,9.5,10. So the right tail is longer, meaning the distribution is right - skewed? Wait, no, left - skewed is when the left tail is longer. Wait, maybe I made a mistake. Let's recall: left - skewed (negative skew) has a long left tail, right - skewed (positive skew) has a long right tail. In our data, the larger values (10,9.5,9) are further out on the right, so the distribution is right - skewed? Wait, but the option says "left - skewed". Wait, maybe I sorted wrong. Wait the original data: 5.5,6,7,10,7.5,8,9.5,9,8.5,8,7,7.5,6,6.5,5.5. Wait, 10 is an outlier on the right. So the mean is pulled up by 10, 9.5, 9. So mean (≈7.43) is less than median (7.5)? Wait no, 7.43 is less than 7.5? Wait 7.43 < 7.5. Wait, if mean < median, the distribution is left - skewed (because the mean is pulled down by low values, but wait no: left - skewed: mean < median, right - skewed: mean > median. Wait, let's check the formula. In left - skewed, the tail is on the left (low values), so mean is less than median. In right - skewed, tail on the right (high values), mean is greater than median. Wait our mean is ≈7.43, median is 7.5. So mean < median, so the distribution is left - skewed? Wait that contradicts the earlier thought. Wait let's take a simple example: if we have data 1,2,3,4,10. Median is 3, mean is (1+2+3+4+10)/5 = 20/5 = 4. So mean > median, right - skewed (tail on right). If data is 1,5,6,7,8. Median 6, mean (1+5+6+7+8)/5 = 27/5 = 5.4. Mean < median, left - skewed (tail on left). Ah, so in our case, mean (7.43) < median (7.5), so the distribution is left - skewed? Wait but our data has a tail on the right (10,9.5,9), but the mean is less than median. Wait maybe the low values (5.5,5.5,6,6) are pulling the mean down. Let's see: the two 5.5s, two 6s: these are low values. So even though there are high values on the right, the low values on the left are more in number? Wait no, the number of low values: 5.5 (2), 6 (2), 6.5 (1) → 5 values. High values: 8.5 (1), 9 (1), 9.5 (1), 10 (1) → 4 values. Middle values: 7 (2), 7.5 (2), 8 (2) → 6 values. So the low values (5.5,6,6.5) are 5 values, high values (8.5,9,9.5,10) are 4 values. So the left tail (low values) has more data points? Wait no, left tail is the extreme low values. The two 5.5s are extreme low, two 6s are low, 6.5 is moderate low. The right tail: 8.5 is moderate high, 9,9.5,10 are extreme high. So even though there are more low - moderate values, the extreme low (5.5) and extreme high (10,9.5,9) are the tails. Wait, maybe the key is mean vs median. Since mean < median, the distribution is left - skewed. So the statement "The histogram will have a shape that is left - skewed" is true? Wait but let's check the options. Wait the second option: "The histogram will show that the mean time is greater than the median time of 7.4 minutes." Wait our median is 7.5, not 7.4. Wait the option says "median time of 7.4 minutes" which is wrong, because we calculated median as 7.5. Wait maybe a typo, but our median is 7.5. The first option: "A histogram will show that the mean time is approximately equal to the median time of 7.5 minutes." Since mean ≈7.43 ≈7.5, this is true. The third option: "The shape of the histogram can be approximated with a normal curve." Normal curve is symmetric, but our distribution is left - skewed (mean < median), so it's not symmetric, so this is false. The fourth option: "The histogram will show that most of the data is centered between 6 minutes and 9 minutes." Let's see the data: 5.5,5.5,6,6,6.5,7,7,7.5,7.5,8,8,8.5,9,9.5,10. The data from 6 to 9: 6,6,6.5,7,7,7.5,7.5,8,8,8.5,9 → that's 11 out of 15 data points. So most of the data is between 6 and 9, so this is true. Wait, but earlier we thought the distribution is left - skewed, but maybe the "most of the data between 6 and 9" is true. Wait let's re - evaluate the options:

1. "A histogram will show that the mean time is approximately equal to the median time of 7.5 minutes." → Mean ≈7.43 ≈7.5, median is 7.5. So this is true.
2. "The histogram will have a shape that is left - skewed." → Mean < median, so left - skewed. Let's check the sorted data: the left has more low - valued data points (5.5,5.5,6,6,6.5) and the right has high - valued but fewer. So the tail is on the left (since mean is pulled down by left values), so left - skewed. So this is true? Wait but when we look at the data, the right has 10,9.5,9 which are outliers, but the left has 5.5,5.5,6,6 which are also outliers. Wait maybe the key is mean < median → left - skewed. So this could be true. But wait our earlier calculation of mean was 111.5 / 15 ≈7.43, median 7.5. So mean < median → left - skewed. So this statement is true?
3. "The histogram will show that the mean time is greater than the median time of 7.4 minutes." → Median is 7.5, not 7.4, and mean (7.43) < median (7.5), so this is false.
4. "The shape of the histogram can be approximated with a normal curve." → Normal curve is symmetric, our distribution is left - skewed, so false.
5. "The histogram will show that most of the data is centered between 6 minutes and 9 minutes." → As we saw, 11 out of 15 data points are between 6 and 9 (inclusive), so this is true.

Wait, but the problem says "Select three options". Wait maybe I made a mistake in the skew. Let's re - calculate the mean:

Data points: 5.5, 5.5, 6, 6, 6.5, 7, 7, 7.5, 7.5, 8, 8, 8.5, 9, 9.5, 10.

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

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 = 111.5 / 15 ≈7.433...

Median: 15 data points, median is the 8th value. Sorted data:

1:5.5, 2:5.5, 3:6, 4:6, 5:6.5, 6:7, 7:7, 8:7.5, 9:7.5, 10:8, 11:8, 12:8.5, 13:9, 14:9.5, 15:10.

So median is 7.5.

Now, let's analyze each option:

- **Option 1: "A histogram will show that the mean time is approximately equal to the median time of 7.5 minutes."** → Mean ≈7.43 ≈7.5, median is 7.5. So this is true.
- **Option 2: "The histogram will have a shape that is left - skewed."** → In a left - skewed distribution, the mean is less than the median (because the left tail (low values) pulls the mean down). Here, mean (≈7.43) < median (7.5), so the distribution is left - skewed. So this is true.
- **Option 3: "The histogram will show that the mean time is greater than the median time of 7.4 minutes."** → Median is 7.5, not 7.4, and mean (≈7.43) < median (7.5). So this is false.
- **Option 4: "The shape of the histogram can be approximated with a normal curve."** → A normal curve is symmetric, but our distribution is left - skewed (mean ≠ median, and the data is not symmetric). So this is false.
- **Option 5: "The histogram will show that most of the data is centered between 6 minutes and 9 minutes."** → Let's count the number of data points between 6 and 9 (inclusive):

- 6: 2, 6.5:1, 7:2, 7.5:2, 8:2, 8.5:1, 9:1.

- Total: 2 + 1+2 + 2+2 + 1+1 = 11. Out of 15 data points, 11 are between 6 and 9. So most of the data is centered between 6 and 9. So this is true.

Wait, but the problem says "Select three options". So the three true options are:

1. A histogram will show that the mean time is approximately equal to the median time of 7.5 minutes.

2. The histogram will have a shape that is left - skewed.

5. The histogram will show that most of the data is centered between 6 minutes and 9 minutes.

But wait, let's check the option about the normal curve again. A normal curve is symmetric, and our data has a left - skew, so it can't be approximated by a normal curve. So option 4 is false. The option "The histogram will show that the mean time is greater than the median time of 7.4 minutes" is false because median is 7.5 and mean is less than median.

So the three true options are:

- 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 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 histogram will have a shape that is left - skewed.
• The histogram will show that most of the data is centered between 6 minutes and 9 minutes.