\begin{tabular}{l} Warm-Up Activo \ $$ \begin{array}{l} ext { Analyzing Average Wind Patterns } \ ext { The monthly wind speeds over a one-year period at } \ ext { Denver International Airport were recorded and the } \ ext { values for each month averaged. The average monthly } \ ext { wind speeds, in mph, from January to December during } \ ext { that time period were } 9.7,10.0,10.8,11,9,11.0,10.7,\end{array} $$ \ $$ \begin{array}{l} ext { Use the statistics calculator to find the statistical } \ ext { measures. } \ ext { The median of the data set is } \ ext { The mean of the data set is }\end{array} $$ \ $$ \begin{array}{l} ext { The population standard deviation of the data set is }\end{array} $$ \ \hline\end{tabular}

Question

\begin{tabular}{l} Warm-Up Activo \ $$ \begin{array}{l}	ext { Analyzing Average Wind Patterns } \ 	ext { The monthly wind speeds over a one-year period at } \ 	ext { Denver International Airport were recorded and the } \ 	ext { values for each month averaged. The average monthly } \ 	ext { wind speeds, in mph, from January to December during } \ 	ext { that time period were } 9.7,10.0,10.8,11,9,11.0,10.7,\end{array} $$ \ $$ \begin{array}{l}	ext { Use the statistics calculator to find the statistical } \ 	ext { measures. } \ 	ext { The median of the data set is } \ 	ext { The mean of the data set is }\end{array} $$ \ $$ \begin{array}{l}	ext { The population standard deviation of the data set is }\end{array} $$ \ \hline\end{tabular}

UpStudy AI · Accepted Answer

Certainly! Let's analyze the provided wind speed data to find the **median**, **mean**, and **population standard deviation**.

### Given Data:
Monthly wind speeds (in mph):  
9.7, 10.0, 10.8, 11, 9, 11.0, 10.7

### 1. **Median of the Data Set**

**Step 1:** Arrange the data in ascending order:
$$ 9, \, 9.7, \, 10.0, \, 10.7, \, 10.8, \, 11, \, 11.0 $$

**Step 2:** Since there are **7** data points (an odd number), the median is the middle value.

- **Median:** **10.7 mph**

### 2. **Mean of the Data Set**

**Step 1:** Sum all the wind speed values.
$$
9.7 + 10.0 + 10.8 + 11 + 9 + 11.0 + 10.7 = 72.2 \, \text{mph}
$$

**Step 2:** Divide the sum by the number of data points.
$$
\text{Mean} = \frac{72.2}{7} \approx 10.31 \, \text{mph}
$$

- **Mean:** **10.31 mph**

### 3. **Population Standard Deviation of the Data Set**

**Step 1:** Calculate each data point's deviation from the mean and square it.

| Data Point | Deviation from Mean ($$x_i - \mu$$) | Squared Deviation $$(x_i - \mu)^2$$ |
|------------|-------------------------------------|------------------------------------|
| 9          | $$9 - 10.31 = -1.31$$               | $$1.71$$                           |
| 9.7        | $$9.7 - 10.31 = -0.61$$             | $$0.37$$                           |
| 10.0       | $$10.0 - 10.31 = -0.31$$            | $$0.10$$                           |
| 10.7       | $$10.7 - 10.31 = 0.39$$             | $$0.15$$                           |
| 10.8       | $$10.8 - 10.31 = 0.49$$             | $$0.24$$                           |
| 11         | $$11 - 10.31 = 0.69$$                | $$0.47$$                           |
| 11.0       | $$11.0 - 10.31 = 0.69$$              | $$0.47$$                           |

**Step 2:** Sum all squared deviations.
$$
1.71 + 0.37 + 0.10 + 0.15 + 0.24 + 0.47 + 0.47 = 3.51
$$

**Step 3:** Calculate the variance (since it's a population standard deviation, we divide by $$N$$, not $$N-1$$).
$$
\text{Variance} = \frac{3.51}{7} \approx 0.50
$$

**Step 4:** Take the square root of the variance to find the standard deviation.
$$
\text{Population Standard Deviation} = \sqrt{0.50} \approx 0.71 \, \text{mph}
$$

- **Population Standard Deviation:** **0.71 mph**

### **Summary:**
- **Median:** 10.7 mph
- **Mean:** 10.31 mph
- **Population Standard Deviation:** 0.71 mph
- **Median:** 10.7 mph - **Mean:** 10.31 mph - **Population Standard Deviation:** 0.71 mph

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