https://file.aiquickdraw.com/mathbot/file/1729936830869otma6abk.jpg

To help with the problem, I'll describe the steps based on what the image displays:

### (a) Choose the correct time series plot:
The problem shows four graphs representing the time series for the cost per unit over time (quarters). You need to match the correct graph to the data table provided. Based on the patterns visible in each plot, select the one that accurately represents the cost data trend over time.

### (b) Describe the pattern in the dataTo help with the problem, I'll describe the steps based on what the image displays:

### (a) Choose the correct time series plot:
The problem shows four graphs representing the time series for the cost per unit over time (quarters). You need to match the correct graph to the data table provided. Based on the patterns visible in each plot, select the one that accurately represents the cost data trend over time.

### (b) Describe the pattern in the 
Here you need to identify the pattern: whether it's *linear*, *quadratic*, *exponential*, or another type. Choose the pattern and how well it fits the data (e.g., fits well or occasionally).

### (c) Determine the linear regression equation:
Calculate the equation of the line that best fits the time series data. Use the formula:
\[
y = mx + b
\]
where \( y \) is the cost per unit, \( m \) is the slope, \( x \) is the time (quarters), and \( b \) is the y-intercept. Then, find the Mean Squared Error (MSE) for this fit.

### (d) Average increase in cost:
Compute the average annual increase in the cost. This can be derived from the slope of the linear regression. Since the data is given in quarters, convert it to an annual increase by multiplying the slope by 4 (quarters per year).

### (e) Estimate the cost for the next period:
Using the linear regression equation, predict the cost for the next quarter (extend the timeline by 1 quarter).

To proceed, please confirm if you want detailed step-by-step solutions for each part or clarification on any specific section. Here are some relative questions to consider:

1. What type of pattern would you expect if the data is generally increasing?
2. How can you interpret the slope in the context of manufacturing costs?
3. What does a low Mean Squared Error indicate about a model's fit?
4. How do you convert quarterly data to annual predictions?
5. What other methods besides linear regression could fit time series data?

**Tip:** Always visualize your data before selecting a fitting model to identify possible patterns or anomalies.

Explore a detailed solution for analyzing a time series in manufacturing cost over multiple years. Learn to choose the correct time series plot, identify data patterns, calculate a linear regression model, determine Mean Squared Error, compute annual cost increase, and forecast future values. Suitable for Grades 11-12.

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