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Let's work through the given problem step-by-step:

### Part (a): Determine the Line of Best Fit Using Linear Regression
We have the following data points for the years and the number of sneezes:

| Year (x) | Sneezes (y) |
|----------|-------------|
| 2018     | 234         |
| 2019     | 243         |
| 2020     | 255         |
| 2021     | 264         |

We will use linear regression to find the line of best fit, which has the general equation:
\[
y = mx + b
\]
Where:
- \( m \) is the slope of the line.
- \( b \) is the y-intercept.

To find \( m \) and \( b \), we need to compute the following sums:
- \( \sum x \): Sum of x-values (years).
- \( \sum y \): Sum of y-values (sneezes).
- \( \sum xy \): Sum of the product of corresponding x and y values.
- \( \sum x^2 \): Sum of the square of x-values.

After calculating these, we can use the formulas:
\[
m = \frac{N\sum xy - (\sum x)(\sum y)}{N\sum x^2 - (\sum x)^2}
\]
\[
b = \frac{\sum y - m\sum x}{N}
\]
Where \( N \) is the number of data points (in this case, 4).

### Part (b): Estimate Sneezes in 2017 Using the Line of Best Fit
Once we have the equation of the line from part (a), substitute \( x = 2017 \) into the equation to estimate the number of sneezes.

### Part (c): Estimate the Year with 200 Sneezes Using the Line of Best Fit
Set \( y = 200 \) in the line equation and solve for \( x \) (year).

---

I'll compute these values for you.### Solutions:
(a) **Line of Best Fit**:  
The equation of the line of best fit using linear regression is:
\[
y = 10.20x - 20349.90
\]

(b) **Estimate Sneezes in 2017**:  
Using the line of best fit, the estimated number of times sneezed in 2017 is:
\[
y(2017) = 10.20(2017) - 20349.90 = 223.5
\]

(c) **Estimate the Year with 200 Sneezes**:  
Setting \( y = 200 \) in the line equation, the estimated year when 200 sneezes occurred is:
\[
x = \frac{200 + 20349.90}{10.20} \approx 2014.7
\]
This corresponds to approximately the year 2015.

Let me know if you have questions or need more detailed explanations.

### Related Questions:
1. How do we determine the goodness of fit for a linear regression model?
2. What other models could be used for fitting if the data is not linear?
3. How can we interpret the slope and intercept in a real-world context?
4. What would happen if we used more data points to determine the line of best fit?
5. How do outliers affect the linear regression line?

**Tip:** When using linear regression, always check if the model assumptions hold true, such as linearity, independence, and homoscedasticity.

The following data describes the number of times I sneezed (y) each year (x). a) Determine the line of best fit using a linear regression. b) Using the line of best fit, estimate how many times I sneezed in 2017. c) Using the line of best fit, estimate what year I sneezed 200 times.

Learn how to determine the line of best fit using linear regression for sneezing data. This problem covers linear regression formulas, slope, and intercept calculations, and provides estimations for sneezing frequency in specific years.

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