Listed below are the amounts​ (dollars) it costs for marriage proposal packages at different baseball stadiums. Find the​ range, variance, and standard deviation for the given sample data. Include appropriate units in the results. Are there any​ outliers, and are they likely to have much of an effect on the measures of​ variation? 39

45

50

65

70

85

85

170

195

218

260

350

500

1500

2750

To analyze the given data, we'll calculate the range, variance, and standard deviation for the marriage proposal package costs at different baseball stadiums. We'll also determine if there are any outliers and assess their potential effect on the measures of variation.

### 1. **Range**:
The range is the difference between the largest and smallest values in the dataset.

\[
\text{Range} = \text{Maximum Value} - \text{Minimum Value}
\]

Given the dataTo analyze the given data, we'll calculate the range, variance, and standard deviation for the marriage proposal package costs at different baseball stadiums. We'll also determine if there are any outliers and assess their potential effect on the measures of variation.

### 1. **Range**:
The range is the difference between the largest and smallest values in the dataset.

\[
\text{Range} = \text{Maximum Value} - \text{Minimum Value}
\]

Given the  39, 45, 50, 65, 70, 85, 85, 170, 195, 218, 260, 350, 500, 1500, 2750

\[
\text{Range} = 2750 - 39 = 2711
\]

Thus, the range is **2711 dollars**.

### 2. **Variance**:
Variance measures the average squared deviation from the mean.

First, find the mean $ \mu $ of the dataTo analyze the given data, we'll calculate the range, variance, and standard deviation for the marriage proposal package costs at different baseball stadiums. We'll also determine if there are any outliers and assess their potential effect on the measures of variation.

### 1. **Range**:
The range is the difference between the largest and smallest values in the dataset.

\[
\text{Range} = \text{Maximum Value} - \text{Minimum Value}
\]

Given the  39, 45, 50, 65, 70, 85, 85, 170, 195, 218, 260, 350, 500, 1500, 2750

\[
\text{Range} = 2750 - 39 = 2711
\]

Thus, the range is **2711 dollars**.

### 2. **Variance**:
Variance measures the average squared deviation from the mean.

First, find the mean $ \mu $ of the 

\[
\mu = \frac{\sum \text{data points}}{\text{number of data points}} = \frac{39 + 45 + 50 + 65 + 70 + 85 + 85 + 170 + 195 + 218 + 260 + 350 + 500 + 1500 + 2750}{15}
\]

\[
\mu = \frac{6382}{15} \approx 425.47
\]

Now, calculate the variance $ \sigma^2 $:

\[
\sigma^2 = \frac{\sum (x_i - \mu)^2}{n-1}
\]
Where $ x_i $ is each data point and $ n $ is the number of data points (15 in this case).

### 3. **Standard Deviation**:
The standard deviation is the square root of the variance:

\[
\sigma = \sqrt{\sigma^2}
\]

Let me compute these values for you now.### Calculations:

1. **Range**: The range is **2711 dollars**.

2. **Variance**: The variance is approximately **548,667.27 dollars²**.

3. **Standard Deviation**: The standard deviation is approximately **740.72 dollars**.

### Outliers:
The value **2750** seems significantly higher than the others, indicating it could be an outlier. 

#### Impact of Outliers:
The outlier (2750) likely has a substantial effect on the variance and standard deviation, inflating these measures. Without this extreme value, the dataset would show much lower variation.

Would you like further details on calculating outliers or removing the outlier to compare?

---

Here are 5 related questions for you:

1. What is the median cost of these packages?
2. How would removing the outlier affect the mean?
3. Can you calculate the interquartile range (IQR) for this data?
4. What is the effect of outliers on the mean versus the median?
5. How could we use the standard deviation to identify potential outliers?

**Tip**: Variance gives insight into how spread out the data is from the mean, while standard deviation directly relates this to the original units of the data.

Listed below are the amounts​ (dollars) it costs for marriage proposal packages at different baseball stadiums. Find the​ range, variance, and standard deviation for the given sample data. Include appropriate units in the results. Are there any​ outliers, and are they likely to have much of an effect on the measures of​ variation? 39, 45, 50, 65, 70, 85, 85, 170, 195, 218, 260, 350, 500, 1500, 2750.

Discover how to calculate the range, variance, and standard deviation for marriage proposal packages at baseball stadiums, with costs ranging from $39 to $2750. This example demonstrates key statistical concepts such as measures of spread and outliers, suitable for Grades 10-12 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation