Pain: The General Social Survey asked  people how many days they would wait to seek medical treatment if they were suffering pain that interfered with their ability to work. The results are presented in the following table.
Number of Days	Frequency
TotalPain: The General Social Survey asked 

823

 people how many days they would wait to seek medical treatment if they were suffering pain that interfered with their ability to work. The results are presented in the following table.


**Number of DaysFrequency**

0

27

1

443

2

257

3

70

4

12

5

14

Total 823 Send data to Excel Consider these 823 people to be a population. Let *X* be the number of days for a person sampled at random from this population.



Let's analyze the given data to understand the problem and calculate relevant statistics.

### Data Breakdown:
- **Number of Days (X):** The number of days a person would wait before seeking medical treatment.
- **Frequency:** The number of people who chose each specific number of days.

### Frequency Distribution:
| **Number of Days (X)** | **Frequency (f)** |
|------------------------|-------------------|
| 0                      | 27                |
| 1                      | 443               |
| 2                      | 257               |
| 3                      | 70                |
| 4                      | 12                |
| 5                      | 14                |
| **Total**              | **823**           |

### Step-by-Step Calculation:
1. **Mean (Expected Value \( E(X) \)):**
   \[
   E(X) = \frac{\sum (X \times f)}{N}
   \]
   where \( N \) is the total number of observations (823).

2. **Variance \( \sigma^2 \):**
   \[
   \sigma^2 = \frac{\sum (X^2 \times f)}{N} - \left(E(X)\right)^2
   \]

3. **Standard Deviation \( \sigma \):**
   \[
   \sigma = \sqrt{\sigma^2}
   \]

Let’s start by calculating the mean \( E(X) \).

### Calculation of the Mean:
First, compute \( X \times f \) for each day.

| **Number of Days (X)** | **Frequency (f)** | **\( X \times f \)** |
|------------------------|-------------------|----------------------|
| 0                      | 27                | 0                    |
| 1                      | 443               | 443                  |
| 2                      | 257               | 514                  |
| 3                      | 70                | 210                  |
| 4                      | 12                | 48                   |
| 5                      | 14                | 70                   |
| **Total**              | **823**           | **1285**             |

So, the mean \( E(X) \) is:
\[
E(X) = \frac{1285}{823} \approx 1.561
\]

Now, let’s calculate the variance and standard deviation. We'll calculate \( X^2 \times f \) next.### Results:
1. **Mean (Expected Value \( E(X) \)):** \( \approx 1.561 \) days
2. **Variance \( \sigma^2 \):** \( \approx 0.774 \) days²
3. **Standard Deviation \( \sigma \):** \( \approx 0.880 \) days

These values summarize the distribution of the number of days people would wait before seeking medical treatment.

Would you like more details or have any specific questions about this analysis?

### Follow-up Questions:
1. How would the mean change if more people waited exactly 0 days?
2. What is the probability that a randomly chosen person would wait exactly 1 day?
3. How would the variance change if everyone waited exactly the same number of days?
4. What does the standard deviation tell us about the population's waiting behavior?
5. How could we calculate the probability distribution from this data?
6. What if we wanted to find the probability of waiting 2 or more days?
7. How would removing outliers affect the mean and standard deviation?
8. How does this distribution compare to a normal distribution?

**Tip:** Variance and standard deviation are crucial for understanding how spread out the data is around the mean.

Explore the statistical analysis of waiting periods before seeking medical treatment for pain interference with work. Calculate the mean, variance, and standard deviation using data from the General Social Survey. Understand the distribution and implications of waiting behaviors.

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