For some reason the quality of production decreases as the year progresses at a light bulb manufacturing plant.  The following data represent the percentage of defective light bulbs produced at a light bulb manufacturing plant in the corresponding month of the year.
Start 2 By 8 Table 1st Row 1st Column month left parenthesis x right parenthesis 2nd Column 2 3rd Column 3 4st Column 5 5st Column 7 6st Column 8 7st Column 9 8st Column 12 2nd Row 1st Column % defective left parenthesis y right parenthesis 2nd Column 1.3 3rd Column 1.6 4st Column 2.0 5st Column 2.4 6st Column 2.6 7st Column 2.8 8st Column 3.1 EndTable 
 
Use the equation of the least squares line to predict the percentage of defective bulbs in June.

To find the least squares regression line, we need to use the following formula for the line \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

The slope \( m \) can be calculated by:

\[
m = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{n(\Sigma x^2) - (\Sigma x)^2}
\]

The intercept \( b \) can be calculated by:

\[
b = \frac{(\Sigma y)(\Sigma x^2) - (\Sigma x)(\Sigma xy)}{n(\Sigma x^2) - (\Sigma x)^2}
\]

Here are the data points given in the table:

| Month (x) | 2  | 3  | 5  | 7  | 8  | 9  | 12 |
|-----------|----|----|----|----|----|----|----|
| % Defective (y) | 1.3 | 1.6 | 2.0 | 2.4 | 2.6 | 2.8 | 3.1 |

### Step 1: Summing values
We will need to compute the following values:

- \( \Sigma x \) (sum of months)
- \( \Sigma y \) (sum of percentages defective)
- \( \Sigma xy \) (sum of products of x and y)
- \( \Sigma x^2 \) (sum of squares of x)

Let me calculate these first.The equation of the least squares regression line is:

\[
y = 0.184x + 1.047
\]

### Step 2: Predict the percentage of defective bulbs in June (month 6)

To predict the percentage of defective bulbs in June, substitute \( x = 6 \) into the equation:

\[
y = 0.184(6) + 1.047
\]

Let's compute the predicted value.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like me to explain the steps for solving this manually, or do you have any other questions?

Here are 5 related questions to deepen your understanding:
1. How is the least squares method used to minimize prediction errors?
2. What are the benefits of using regression analysis in quality control?
3. Can you calculate residuals for the data points and analyze the model's accuracy?
4. How can you interpret the slope in the context of defective bulbs?
5. How might external factors influence the trend of defective bulbs over time?

**Tip:** To improve accuracy in predictions, ensure your data points are spread out and representative of various scenarios.

For some reason the quality of production decreases as the year progresses at a light bulb manufacturing plant. The following data represent the percentage of defective light bulbs produced at a light bulb manufacturing plant in the corresponding month of the year. Start 2 By 8 Table 1st Row 1st Column month left parenthesis x right parenthesis 2nd Column 2 3rd Column 3 4st Column 5 5st Column 7 6st Column 8 7st Column 9 8st Column 12 2nd Row 1st Column % defective left parenthesis y right parenthesis 2nd Column 1.3 3rd Column 1.6 4st Column 2.0 5st Column 2.4 6st Column 2.6 7st Column 2.8 8st Column 3.1 EndTable. Use the equation of the least squares line to predict the percentage of defective bulbs in June.

Learn how to use the least squares regression line to predict the percentage of defective light bulbs produced at a light bulb manufacturing plant. This problem involves calculating the regression line from given data points and using it to make predictions, suitable for students in Grades 10-12.

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