An article on pedaling techniques for elite endurance cyclists reported on single-leg power at a high workload.

247

193

160

184

178

178

177

204

211

190

214

185

199

197

(a) Calculate the sample mean *x* and median . (Round your mean to two decimal places.)

*x*

=  

=  


Interpret the sample mean and median.

The mean is larger than the median, but they are still fairly close together.The mean is much larger than the median.    The median is much larger than the mean.The median is larger than the mean, but they are still fairly close together.


(b) Suppose that the first observation had been 254 rather than 247. Calculate the sample mean and median. (Round your mean to two decimal places.)

*x*

=  

=  


How would the mean and median change?

Both the mean and median stayed the same.Both the mean and median decreased.    The mean decreased, and the median stayed the same.The mean increased, and the median stayed the same.


(c) Calculate a trimmed mean by eliminating the smallest and largest sample observations. (Enter your answer to two decimal places.)
 

What is the corresponding trimming percentage? (Round your answer to two decimal places.)
  %

Let's break this problem into parts and solve it step-by-step:

### (a) Calculate the sample mean and median.

We are given the following dataset:

\[
247, 193, 160, 184, 178, 178, 177, 204, 211, 190, 214, 185, 199, 197
\]

1. **Sample Mean**:
The sample mean \(\bar{x}\) is calculated by summing all the values and dividing by the total number of observations. 

The formula for the mean is:
\[
\bar{x} = \frac{\sum x_i}{n}
\]

where \(x_i\) represents each data point and \(n\) is the number of data points.

2. **Median**:
The median is the middle value in a sorted dataset. If the number of observations is even, the median is the average of the two middle numbers.

Let's compute these values.

The sample mean is approximately \( \bar{x} = 194.07 \), and the median is \( 191.5 \).

### Interpretation of the Mean and Median
Since the mean is slightly larger than the median, but they are still fairly close together, the correct interpretation is:

- **"The mean is larger than the median, but they are still fairly close together."**

---

### (b) Calculate the sample mean and median if the first observation had been 254 instead of 247.

Let's update the dataset and calculate the new mean and median.It seems like I can’t do more advanced data analysis right now. Please try again later.

If you'd like, you can follow these steps to calculate the new mean and median manually:

1. **New Dataset**: Replace the first value (247) with 254 in the original dataset:
   \[
   254, 193, 160, 184, 178, 178, 177, 204, 211, 190, 214, 185, 199, 197
   \]

2. **New Mean**:
   \[
   \text{Mean} = \frac{\text{Sum of values}}{14}
   \]

3. **New Median**: Sort the updated dataset and find the middle values. Since there are 14 data points, average the 7th and 8th values after sorting.

Would you like further assistance with the next steps?

Here are 5 related questions:
1. What is the impact of outliers on the mean and median?
2. How do you calculate a trimmed mean, and why is it useful?
3. What does the median tell you about the distribution of data?
4. How does increasing a single data point affect the standard deviation?
5. Why are the mean and median close together in symmetrical distributions?

**Tip**: The median is less sensitive to extreme values (outliers) than the mean.

An article on pedaling techniques for elite endurance cyclists reported on single-leg power at a high workload.

247
193
160
184
178
178
177
204
211
190
214
185
199
197
(a) Calculate the sample mean *x* and median. (Round your mean to two decimal places.)

*x* =

Interpret the sample mean and median. The mean is larger than the median, but they are still fairly close together.

(b) Suppose that the first observation had been 254 rather than 247. Calculate the sample mean and median. (Round your mean to two decimal places.)

*x* =

How would the mean and median change?

(c) Calculate a trimmed mean by eliminating the smallest and largest sample observations.

What is the corresponding trimming percentage?

Learn how to calculate the sample mean, median, and trimmed mean using cyclists' power data. This problem involves key statistical concepts such as measures of central tendency and demonstrates the effects of outliers on the mean and median. Suitable for students in Grades 10-12.

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