Capacity and price data for a collection of USB flash drives is displayed in the table below.



Capacity (Gb)Price ($)

8

9.99

16

12.99

32

16.99

64

27.99

128

48.99

256

89.99


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.

y^=


To find the best-fit line for the data using linear regression, we will calculate the equation of the form:

\[
\hat{y} = mx + b
\]

where:
- $x$ is the capacity (in Gb),
- $\hat{y}$ is the predicted price (in $),
- $m$ is the slope of the line,
- $b$ is the y-intercept.

The general formula for the slope $m$ and y-intercept $b$ in linear regression are:

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

\[
b = \frac{\sum y - m(\sum x)}{n}
\]

Where $n$ is the number of data points, and the sums refer to summations over all the data points. Let's calculate it using the provided dataTo find the best-fit line for the data using linear regression, we will calculate the equation of the form:

\[
\hat{y} = mx + b
\]

where:
- $x$ is the capacity (in Gb),
- $\hat{y}$ is the predicted price (in $),
- $m$ is the slope of the line,
- $b$ is the y-intercept.

The general formula for the slope $m$ and y-intercept $b$ in linear regression are:

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

\[
b = \frac{\sum y - m(\sum x)}{n}
\]

Where $n$ is the number of data points, and the sums refer to summations over all the data points. Let's calculate it using the provided 

| Capacity (Gb) $x$ | Price ($) $y$ |
|--------------------|-----------------|
| 8                  | 9.99            |
| 16                 | 12.99           |
| 32                 | 16.99           |
| 64                 | 27.99           |
| 128                | 48.99           |
| 256                | 89.99           |

I'll calculate the values of $m$ and $b$.The equation of the best-fit line for the data, using linear regression, is:

\[
\hat{y} = 0.323x + 7.35
\]

Here:
- The slope $m = 0.323$, meaning that for each additional gigabyte (Gb) of capacity, the price increases by approximately $0.32.
- The y-intercept $b = 7.35$, representing the estimated price when the capacity is 0 Gb.

Would you like further details on how these calculations work? Here are some related questions:

1. How would the regression line look graphically for the given data?
2. How accurate is this linear regression model for predicting prices of larger capacities?
3. What is the significance of the slope in this context?
4. How could we use this equation to predict the price of a 512 Gb USB drive?
5. How would the regression change if an additional price-capacity pair was added to the data?

**Tip:** Always check how well a regression model fits the data by analyzing the residuals or calculating $R^2$, the coefficient of determination.

Capacity and price data for a collection of USB flash drives is displayed in the table below. 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).

Calculate the best-fit line using linear regression for USB flash drives' capacity and price data. This problem involves statistical concepts and linear regression formulas to determine the relationship between capacity and price. Suitable for students in Grades 10-12.

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