A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN: 9780134753119
Author: Sheldon Ross
Publisher: PEARSON
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**Hours Studied and Test Grades**

The following table shows the hours studied and corresponding test grade earned by students on a recent test. Calculate the correlation coefficient, \(r\), and determine whether \(r\) is statistically significant at the 0.01 level of significance. Round your answer to the nearest thousandth.

**Critical Values of the Pearson Correlation Coefficient**

| Hours Studied and Test Grades |
|-------------------------------|
| **Hours Studied** | 0 | 0.5 | 1 | 1 | 2 | 2.25 | 3.25 | 3.5 | 3.75 | 3.75 | 4 | 5 |
| **Test Grade** | 72 | 64 | 68 | 81 | 79 | 85 | 82 | 72 | 99 | 87 | 84 | 97 |
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Transcribed Image Text:**Hours Studied and Test Grades** The following table shows the hours studied and corresponding test grade earned by students on a recent test. Calculate the correlation coefficient, \(r\), and determine whether \(r\) is statistically significant at the 0.01 level of significance. Round your answer to the nearest thousandth. **Critical Values of the Pearson Correlation Coefficient** | Hours Studied and Test Grades | |-------------------------------| | **Hours Studied** | 0 | 0.5 | 1 | 1 | 2 | 2.25 | 3.25 | 3.5 | 3.75 | 3.75 | 4 | 5 | | **Test Grade** | 72 | 64 | 68 | 81 | 79 | 85 | 82 | 72 | 99 | 87 | 84 | 97 |
### Statistical Significance of Pearson's Correlation Coefficient

#### Determine the Significance of \[ r \]

\[ r = \_\_\_\_\_\_\_\_\_ \]

Is \[ r \] statistically significant at the 0.01 level of significance?

**Instructions:**

1. **Enter the value of \[ r \]**: Input the calculated Pearson's correlation coefficient value in the provided space.
2. **Evaluate Significance**: Use statistical tables or software to determine if the value of \[ r \] is significant at the 0.01 level. The significance indicates how likely it is that the observed correlation is due to chance.

**Step-by-Step Guide:**

1. **Compute the Pearson's correlation coefficient (\[ r \])** using appropriate data.
2. **Refer to the critical value table**: For a given degree of freedom \[ n-2 \], where \[ n \] is the sample size, find the critical value corresponding to the 0.01 significance level.
3. **Compare \[ r \] value**: Check if the absolute value of \[ r \] is greater than the critical value from the table. If it is, then \[ r \] is statistically significant at the 0.01 level.

**Concept Explanation:**

- **Pearson's correlation coefficient (\[ r \])** measures the strength and direction of the linear relationship between two variables.
- **Statistical significance** at the 0.01 level means there is only a 1% chance that the observed correlation occurred randomly, implying a 99% confidence level.

**Further Reading:**

- [Pearson Correlation Coefficient](https://en.wikipedia.org/wiki/Pearson_correlation_coefficient)
- [Significance Testing](https://en.wikipedia.org/wiki/Statistical_significance)

---

This content helps you understand how to determine the statistical significance of a correlation coefficient in your data analysis, ensuring accurate and reliable results.
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Transcribed Image Text:### Statistical Significance of Pearson's Correlation Coefficient #### Determine the Significance of \[ r \] \[ r = \_\_\_\_\_\_\_\_\_ \] Is \[ r \] statistically significant at the 0.01 level of significance? **Instructions:** 1. **Enter the value of \[ r \]**: Input the calculated Pearson's correlation coefficient value in the provided space. 2. **Evaluate Significance**: Use statistical tables or software to determine if the value of \[ r \] is significant at the 0.01 level. The significance indicates how likely it is that the observed correlation is due to chance. **Step-by-Step Guide:** 1. **Compute the Pearson's correlation coefficient (\[ r \])** using appropriate data. 2. **Refer to the critical value table**: For a given degree of freedom \[ n-2 \], where \[ n \] is the sample size, find the critical value corresponding to the 0.01 significance level. 3. **Compare \[ r \] value**: Check if the absolute value of \[ r \] is greater than the critical value from the table. If it is, then \[ r \] is statistically significant at the 0.01 level. **Concept Explanation:** - **Pearson's correlation coefficient (\[ r \])** measures the strength and direction of the linear relationship between two variables. - **Statistical significance** at the 0.01 level means there is only a 1% chance that the observed correlation occurred randomly, implying a 99% confidence level. **Further Reading:** - [Pearson Correlation Coefficient](https://en.wikipedia.org/wiki/Pearson_correlation_coefficient) - [Significance Testing](https://en.wikipedia.org/wiki/Statistical_significance) --- This content helps you understand how to determine the statistical significance of a correlation coefficient in your data analysis, ensuring accurate and reliable results.
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