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:

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.