Part (a): Linear Equation for Immigration

We are asked to assume that the change in immigration is linear. To do this, we will write the equation of the line in slope-intercept form:
$y = mt + b$ where:

• $y$ is the number of immigrants,
• $t$ is the number of years after 1900,
• $m$ is the slope (rate of change of immigration),
• $b$ is the y-intercept (the number of immigrants at $t = 0$, or in the year 1900).

We are given two points in time:

1. In 1960 ($t = 1960 - 1900 = 60$), the number of immigrants was 238,004.
2. In 2007 ($t = 2007 - 1900 = 107$), the number of immigrants was 1,005,190.

Step 1: Find the slope $m$

The slope $m$ is the rate of change of the number of immigrants between 1960 and 2007. Using the formula for the slope between two points: $m = \frac{y_2 - y_1}{t_2 - t_1}$ where:

• $y_2 = 1,005,190$ (number of immigrants in 2007),
• $y_1 = 238,004$ (number of immigrants in 1960),
• $t_2 = 107$ (time in years after 1900 in 2007),
• $t_1 = 60$ (time in years after 1900 in 1960).

Substitute these values: $m = \frac{1,005,190 - 238,004}{107 - 60} = \frac{767,186}{47} \approx 16,325.23$ Thus, the slope $m \approx 16,325.23$.

Step 2: Find the y-intercept $b$

We can now use the slope-intercept form $y = mt + b$ and one of the points (say $t = 60$ and $y = 238,004$) to solve for $b$: $238,004 = 16,325.23(60) + b$ $238,004 = 979,513.8 + b$ $b = 238,004 - 979,513.8 = -741,509.8$ Thus, the y-intercept $b \approx -741,510$.

Final Equation

The equation expressing the number of immigrants $y$ in terms of $t$ is: $y = 16,325.23t - 741,510$

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Part (b): Predicting Immigration in 2016

To predict the number of immigrants in 2016, we substitute $t = 2016 - 1900 = 116$ into the equation we found: $y = 16,325.23(116) - 741,510$ $y = 1,894,127.68 - 741,510 = 1,152,617.68$ Thus, the predicted number of immigrants in 2016 is approximately 1,152,618.

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Part (c): Validity of Using the Equation for the Entire 20th Century

The y-intercept $b \approx -741,510$ suggests that in the year 1900 ($t = 0$), the model predicts a negative number of immigrants. This is not realistic, as immigration numbers cannot be negative. This suggests that while the linear model might fit the data between 1960 and 2007, it is not valid for modeling immigration over the entire 20th century, especially for earlier years.

In real-world scenarios, immigration trends are often influenced by complex factors (such as wars, economic conditions, and policy changes) that cannot be accurately modeled by a simple linear relationship over long periods.

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Additional Questions:

1. How would the equation change if we modeled a different time period, such as from 1980 to 2010?
2. What would be the effect of using an exponential growth model instead of a linear one?
3. Could we model immigration data using a piecewise function to reflect different periods in history more accurately?
4. How would the equation differ if the immigration data showed a sudden increase in a particular year (e.g., due to political changes)?
5. What methods could we use to refine the model to make it more accurate for different time periods?

Tip: Linear models are best for data showing a steady rate of change. For non-linear trends, consider using polynomial or exponential models.