Find the correlation coefficient, r, of the data described below.The design team at an electronics company is evaluating its new prototype for a miniature recording device. As part of this evaluation, designers at the company gathered data about competing devices already on the market.

Among other things, the designers recorded the thickness of each recording device (in millimeters), x, and its maximum recording length (in minutes), y.   

   Thickness (in millimeters)                                      Recording time (in minutes)

                     11                                                                               60

                     21                                                                              60

                     21                                                                               69

                      24                                                                              87

                       29                                                                              93                                           Round* *your answer to the nearest* *thousandth.*r=

To find the correlation coefficient \( r \), we use the formula for Pearson's correlation coefficient:

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

Where:
- \( x \) is the thickness of the devices,
- \( y \) is the recording time,
- \( n \) is the number of data points.

We are given the following dataTo find the correlation coefficient \( r \), we use the formula for Pearson's correlation coefficient:

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

Where:
- \( x \) is the thickness of the devices,
- \( y \) is the recording time,
- \( n \) is the number of data points.

We are given the following 

\[
x: 11, 21, 21, 24, 29
\]
\[
y: 60, 60, 69, 87, 93
\]

First, let's calculate the necessary sums:

- \( \Sigma x \): the sum of the \( x \)-values
- \( \Sigma y \): the sum of the \( y \)-values
- \( \Sigma xy \): the sum of the product of each \( x \) and \( y \)-value
- \( \Sigma x^2 \): the sum of the squares of the \( x \)-values
- \( \Sigma y^2 \): the sum of the squares of the \( y \)-values

Let me calculate that for you.The correlation coefficient \( r \) is approximately 0.819 when rounded to the nearest thousandth. This indicates a strong positive linear relationship between the thickness of the recording devices and their maximum recording time.

Do you want more details or have any questions?

Here are some related questions you might explore:
1. How does the correlation coefficient change if the dataset changes?
2. What does a correlation coefficient of 0 imply about two variables?
3. How would the interpretation differ if \( r \) were negative?
4. Can you predict the recording time for a thickness of 25 mm based on this data?
5. How does the correlation coefficient relate to causality?

**Tip:** Correlation does not imply causation; even a strong correlation like this doesn't mean that one variable causes the other to change.

Find the correlation coefficient, r, of the data described below. The design team at an electronics company is evaluating its new prototype for a miniature recording device. As part of this evaluation, designers at the company gathered data about competing devices already on the market. Among other things, the designers recorded the thickness of each recording device (in millimeters), x, and its maximum recording length (in minutes), y. Thickness (in millimeters): 11, 21, 21, 24, 29. Recording time (in minutes): 60, 60, 69, 87, 93. Round your answer to the nearest thousandth.

Calculate the correlation coefficient, r, between the thickness of recording devices (in millimeters) and their maximum recording time (in minutes). Given data includes device thicknesses of 11, 21, 21, 24, and 29 mm, and recording times of 60, 60, 69, 87, and 93 minutes. Learn how to apply Pearson's correlation coefficient formula step-by-step to find a strong positive relationship between the variables.

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