Let's break this problem down step by step to address each part.

Part (a): Do the data appear linear?

The answer was already provided as Yes, which means that based on the scatter plot, the data seems to follow a linear trend. You were correct here.

Part (b): Find a linear model for the data.

You are asked to find a linear regression model for the data, where:

• $y$ is the value of the dollar in 1982–1984 dollars.
• $t$ is the time variable, where $t = 4$ represents the year 1994.

We need to convert the years into $t$ values. For example:

• 1994 corresponds to $t = 4$
• 1995 corresponds to $t = 5$
• 1996 corresponds to $t = 6$, and so on up to 2008.

Using these points, you can use a spreadsheet software or calculator with a regression feature to find the linear equation. The equation will be of the form:

$y = mt + b$

Where $m$ is the slope and $b$ is the y-intercept. Ensure you round both values to four decimal places.

Part (c): Predict the value of 1 dollar paid by consumers in 2010 and 2011.

• For 2010: $t = 4 + (2010 - 1994) = 4 + 16 = 20$
• For 2011: $t = 4 + (2011 - 1994) = 4 + 17 = 21$

After finding the linear model in part (b), you can substitute $t = 20$ for 2010 and $t = 21$ for 2011 into the equation and solve for $y$.

Part (d): Discuss the reliability of the predictions.

The reliability of predictions depends on whether the scatter plot data closely follows a linear pattern. Since the data appeared linear, the reliability of your predictions depends on how well the linear regression fits the data points.

If the data closely follows a linear pattern:

• The predictions for 2010 and 2011 are reliable.

If the data deviates significantly from a linear pattern:

• The predictions are less reliable.

Based on your initial assumption that the data appears linear, the predictions should be reliable. So the correct statement is:

"Because the data follow a linear pattern, the predictions for 2010 and 2011 are reliable."

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To clarify, you need to:

1. Use the spreadsheet regression feature to find the linear equation.
2. Substitute $t = 20$ and $t = 21$ into the equation to predict the values for 2010 and 2011.
3. Check the scatter plot and regression fit to confirm the reliability of the predictions.

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Do you have any questions or would you like more details on how to calculate the regression?

Follow-up questions:

1. What is the exact formula for calculating the slope and intercept manually for a linear regression model?
2. Can you explain why the linear model might not always fit the data perfectly?
3. What is the difference between interpolation and extrapolation in the context of linear regression predictions?
4. How can residuals be used to check the accuracy of a regression model?
5. Why is it important to round numerical values in regression models?

Tip:

Always visualize your data alongside the regression line. This helps in determining how well the model fits the actual data.