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To solve this problem, we need to obtain the linear trend equation for the data on new checking accounts.

The general form of a linear trend equation is:

\[
Y = a + bt
\]

Where:
- \( Y \) is the number of new accounts predicted,
- \( t \) is the time period,
- \( a \) is the intercept (the value of \( Y \) when \( t = 0 \)),
- \( b \) is the slope (the rate of change of \( Y \) with respect to \( t \)).

### Steps to calculate the linear trend equation:
1. **Identify the time periods (t)**: The periods range from \( t = 1 \) to \( t = 15 \).
2. **Identify the values of \( Y \)**: These are the new accounts corresponding to each period.
3. **Calculate the necessary sums**:
   - Sum of periods (\( \sum t \)),
   - Sum of new accounts (\( \sum Y \)),
   - Sum of the product of periods and new accounts (\( \sum tY \)),
   - Sum of squares of periods (\( \sum t^2 \)).

4. **Use the following formulas to calculate \( a \) and \( b \)**:
   - \( b = \frac{n \sum tY - \sum t \sum Y}{n \sum t^2 - (\sum t)^2} \)
   - \( a = \frac{\sum Y - b \sum t}{n} \)

Where \( n \) is the number of periods (in this case, \( n = 15 \)).

Once we calculate the values of \( a \) and \( b \), we can predict the number of new accounts for periods 16 through 19 using the linear equation.

Let me calculate these values step by step.The linear trend equation based on the data is:

\[
Y = 195.23 + 7.22t
\]

Where:
- \( 195.23 \) is the intercept \( (a) \),
- \( 7.22 \) is the slope \( (b) \).

Now, using this equation, the predicted number of new checking accounts for periods 16 through 19 are:

\[
Y_{16} = 310.77
\]
\[
Y_{17} = 317.99
\]
\[
Y_{18} = 325.21
\]
\[
Y_{19} = 332.44
\]

Let me know if you'd like more details or have any questions.

Here are 5 related questions to expand on this:
1. How would the trend equation change if a different data set were used?
2. What is the significance of the slope in the linear equation in business forecasting?
3. How can we determine if a linear model is the best fit for this data?
4. What are some alternative methods to predict future data points aside from linear regression?
5. How can we interpret the intercept \( a \) in this context?

**Tip:** Always visualize the data to see if a linear model is appropriate before using regression techniques!

Obtain the linear trend equation for the following data on new checking accounts at Fair Savings Bank and use it to predict expected new checking accounts for periods 16 through 19.

Learn how to calculate the linear trend equation for checking accounts at Fair Savings Bank and predict future accounts for periods 16 through 19. This problem involves concepts of linear regression, time series analysis, and business forecasting. Ideal for advanced high school and college students.

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