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

47.99

256

88.99


Use linear regression to find the best-fit line for the data, with capacity as the independent variable and price as the dependent variable. y=

To find the best-fit line using linear regression, we first need to calculate the equation of the line in the form:

\[
y = mx + b
\]

Where:
- $ y $ is the price in dollars,
- $ x $ is the capacity in gigabytes,
- $ m $ is the slope of the line,
- $ b $ is the y-intercept.

### Step 1: Define the data points.
We have the following data pairs:
- (8, 9.99)
- (16, 12.99)
- (32, 16.99)
- (64, 27.99)
- (128, 47.99)
- (256, 88.99)

### Step 2: Calculate the required values.
We need to find the slope $ m $ and intercept $ b $ using the formulas:

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

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

Where:
- $ n $ is the number of data points (6 in this case),
- $ x $ and $ y $ are the values of capacity and price respectively.

### Step 3: Compute the sums.
Let's compute the necessary sums based on the given data.

\[
\sum x = 8 + 16 + 32 + 64 + 128 + 256 = 504
\]

\[
\sum y = 9.99 + 12.99 + 16.99 + 27.99 + 47.99 + 88.99 = 204.94
\]

\[
\sum x^2 = 8^2 + 16^2 + 32^2 + 64^2 + 128^2 + 256^2 = 77440
\]

\[
\sum xy = (8)(9.99) + (16)(12.99) + (32)(16.99) + (64)(27.99) + (128)(47.99) + (256)(88.99) = 31492.8
\]

### Step 4: Substitute the values into the formulas.
Now substitute these sums into the formula for $ m $:

\[
m = \frac{6(31492.8) - (504)(204.94)}{6(77440) - (504)^2}
\]
\[
m = \frac{188956.8 - 103690.56}{464640 - 254016} = \frac{85266.24}{210624} \approx 0.405
\]

Now for $ b $:

\[
b = \frac{(204.94)(77440) - (504)(31492.8)}{6(77440) - (504)^2}
\]
\[
b = \frac{15863705.6 - 15871372.8}{210624} = \frac{-7667.2}{210624} \approx -0.0364
\]

### Step 5: Write the equation.
The equation of the best-fit line is:

\[
y = 0.405x - 0.0364
\]

This is the linear regression model, where $ y $ (price) depends on $ x $ (capacity in GB).

Do you want more details or have any questions?

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Here are 5 related questions:
1. How can we interpret the slope of this best-fit line?
2. What would the price be for a USB flash drive with a capacity of 512 GB using this model?
3. How accurate is this linear model for predicting prices of higher-capacity flash drives?
4. What happens if we include an additional variable, such as brand or speed, in the model?
5. How does the linear regression approach compare to other regression methods, like polynomial regression?

**Tip**: Always check how well the linear regression model fits the data by calculating the R-squared value!

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 and price as the dependent variable.

Learn how to apply linear regression to find the best-fit line for USB flash drive capacity and price data. This problem covers essential algebra and statistics concepts, including the calculation of the slope and y-intercept. Suitable for students in Grades 10-12.

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