https://file.aiquickdraw.com/mathbot/file/1729031040265219f1or4.jpg

Let's work through the problem based on the image you uploaded.

### Step 1: Calculate the correlation coefficient (r)
The correlation coefficient \( r \) can be calculated using the formula:

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

Where:
- \( n \) is the number of data points
- \( x \) and \( y \) are the respective data points for the variables

For the given dataLet's work through the problem based on the image you uploaded.

### Step 1: Calculate the correlation coefficient (r)
The correlation coefficient \( r \) can be calculated using the formula:

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

Where:
- \( n \) is the number of data points
- \( x \) and \( y \) are the respective data points for the variables

For the given 
- \( x \): [3, 4, 5, 6, 7, 8, 9, 10]
- \( y \): [7.4, 11.5, 13.2, 16.6, 17.2, 18.4, 20.2, 20.9]

I will calculate these values for you.

### Step 2: Hypothesis Testing
Next, we'll perform a hypothesis test at a significance level of \( \alpha = 0.01 \).

#### Null hypothesis (\( H_0 \)): \( \rho = 0 \)
There is no significant relationship between the year and the high temperature.

#### Alternative hypothesis (\( H_a \)): \( \rho \neq 0 \)
There is a significant relationship between the year and the high temperature.

For this, we'll:
- Calculate the test statistic \( t \)
- Determine the p-value based on \( t \)

Let me first calculate the correlation coefficient \( r \). I'll work on that.The calculated correlation coefficient \( r = 0.970 \) (rounded to three decimal places).

### Step 2: Testing the Significance of the Correlation

Now, we will test the significance of this correlation coefficient using the given significance level \( \alpha = 0.01 \).

The test statistic \( t \) for testing the correlation is given by:

\[
t = \frac{r \sqrt{n - 2}}{\sqrt{1 - r^2}}
\]

Let me calculate the test statistic \( t \).The calculated test statistic is \( t = 9.778 \) (rounded to three decimal places).

### Step 3: Calculate the p-value

To complete the hypothesis test, we need to calculate the

Calculate the correlation coefficient and perform hypothesis testing on annual high temperatures for a sample of 8 years.

Learn how to calculate the correlation coefficient and perform hypothesis testing on annual high temperatures over a period of 8 years. This exercise covers key statistical concepts such as correlation and hypothesis testing, making it ideal for Grades 10-12 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation