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To find the best-fit line for the given data using linear regression, we need to determine the linear equation $ y = mx + b $ where $ y $ is the price, $ x $ is the capacity, $ m $ is the slope, and $ b $ is the y-intercept. We can calculate the slope and intercept using the formulas for linear regression.

The given data is:

\[
\begin{array}{|c|c|}
\hline
\text{Capacity (Gb)} & \text{Price (\$)} \\
\hline
8 & 9.99 \\
16 & 12.99 \\
32 & 17.99 \\
64 & 26.99 \\
128 & 48.99 \\
256 & 86.99 \\
\hline
\end{array}
\]

### Steps to Calculate Linear Regression:
1. **Calculate the means of $ x $ and $ y $:**

   \[
   \bar{x} = \frac{8 + 16 + 32 + 64 + 128 + 256}{6} = \frac{504}{6} = 84
   \]

   \[
   \bar{y} = \frac{9.99 + 12.99 + 17.99 + 26.99 + 48.99 + 86.99}{6} = \frac{204.94}{6} \approx 34.16
   \]

2. **Calculate the slope $ m $:**

   \[
   m = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}}
   \]

   Let's calculate each term step-by-step.

   \[
   \sum{(x_i - \bar{x})(y_i - \bar{y})} = (8 - 84)(9.99 - 34.16) + (16 - 84)(12.99 - 34.16) + (32 - 84)(17.99 - 34.16) + (64 - 84)(26.99 - 34.16) + (128 - 84)(48.99 - 34.16) + (256 - 84)(86.99 - 34.16)
   \]

   \[
   = (-76)(-24.17) + (-68)(-21.17) + (-52)(-16.17) + (-20)(-7.17) + (44)(14.83) + (172)(52.83)
   \]

   \[
   = 1837.32 + 1439.56 + 841.32 + 143.40 + 652.52 + 9086.36 = 13994.48
   \]

   \[
   \sum{(x_i - \bar{x})^2} = (8 - 84)^2 + (16 - 84)^2 + (32 - 84)^2 + (64 - 84)^2 + (128 - 84)^2 + (256 - 84)^2
   \]

   \[
   = 5776 + 4624 + 2704 + 400 + 1936 + 29412 = 45052
   \]

   \[
   m = \frac{13994.48}{45052} \approx 0.31
   \]

3. **Calculate the intercept $ b $:**

   \[
   b = \bar{y} - m \bar{x} = 34.16 - (0.31 \times 84) \approx 34.16 - 26.04 = 8.12
   \]

Thus, the equation of the best-fit line is:

\[
y = 0.31x + 8.12
\]

Rounded to 2 decimal places, the best-fit line is:

\[
y = 0.31x + 8.12
\]

Would you like more details or have any questions?

Here are 5 related questions to further expand on this topic:
1. How do we interpret the slope and intercept in the context of this problem?
2. What is the correlation coefficient for this data set, and what does it indicate?
3. How would you calculate the predicted price for a USB flash drive with a capacity of 512 GB using the linear regression model?
4. What are the potential limitations of using linear regression for this data?
5. How can you use this linear regression model to make decisions about purchasing USB flash drives?

**Tip:** Always plot your data along with the regression line to visually inspect how well the line fits the data points. This can help in identifying any anomalies or outliers.

Use linear regression to find the best-fit line for the data, with capacity as the independent variable x and price as the dependent variable y. The data provided are Capacity (Gb) and Price ($) for a collection of USB flash drives: (8, 9.99), (16, 12.99), (32, 17.99), (64, 26.99), (128, 48.99), (256, 86.99).

Learn how to apply linear regression to determine the best-fit line for the relationship between the capacity (Gb) and price ($) of USB flash drives. Detailed steps for calculating the slope and intercept are provided, along with relevant formulas. Suitable for Grades 10-12.

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