A data set includes weights of garbage discarded in one week from 62 different households. The paired weights of paper and glass were used to obtain the results shown to the right. Is there sufficient evidence to support the claim that there is a linear correlation between weights of discarded paper and glass? Use a significance level of a = 0.05. Correlation matrix: Variables Paper Glass Click here to view a table of critical values for the correlation coefficient 1l0.1666 Рарer Glass 0.1666 Determine the null and alternative hypotheses Họ: P I (Type integers or decimals. Do not round.) Identify the test statistic, r. r= (Round to three decimal places as needed.) Identify the critical value(s). (Round to three decimal places as needed.) O A. There is one critical value at r= O B. There are two critical values at r= State the conclusion. Because the absolute value of the test statistic is V the positive critical value, there V sufficient evidence to support the claim that there is a linear correlation between the weights of discarded paper and glass for a significance level of a=0.05.

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Certainly! Here is a transcription of the table from the image, which is suitable for an educational website:

---

**Table of Critical Values for Correlation Coefficients**

This table presents critical values for testing the significance of correlation coefficients at two significance levels (\(\alpha = 0.05\) and \(\alpha = 0.01\)). These values indicate the threshold above which a correlation coefficient is considered statistically significant for a given sample size (\(n\)).

| \(n\)  | \(\alpha = 0.05\) | \(\alpha = 0.01\) |
|-------|-----------------|-----------------|
| 4     | 0.950           | 0.990           |
| 5     | 0.878           | 0.959           |
| 6     | 0.811           | 0.917           |
| 7     | 0.754           | 0.875           |
| 8     | 0.707           | 0.834           |
| 9     | 0.666           | 0.798           |
| 10    | 0.632           | 0.765           |
| 11    | 0.602           | 0.735           |
| 12    | 0.576           | 0.708           |
| 13    | 0.553           | 0.684           |
| 14    | 0.532           | 0.661           |
| 15    | 0.514           | 0.641           |
| 16    | 0.497           | 0.623           |
| 17    | 0.482           | 0.606           |
| 18    | 0.468           | 0.590           |
| 19    | 0.456           | 0.575           |
| 20    | 0.444           | 0.561           |
| 25    | 0.396           | 0.505           |
| 30    | 0.361           | 0.463           |
| 35    | 0.335           | 0.430           |
| 40    | 0.312           | 0.402           |
| 45    | 0.294
Transcribed Image Text:Certainly! Here is a transcription of the table from the image, which is suitable for an educational website: --- **Table of Critical Values for Correlation Coefficients** This table presents critical values for testing the significance of correlation coefficients at two significance levels (\(\alpha = 0.05\) and \(\alpha = 0.01\)). These values indicate the threshold above which a correlation coefficient is considered statistically significant for a given sample size (\(n\)). | \(n\) | \(\alpha = 0.05\) | \(\alpha = 0.01\) | |-------|-----------------|-----------------| | 4 | 0.950 | 0.990 | | 5 | 0.878 | 0.959 | | 6 | 0.811 | 0.917 | | 7 | 0.754 | 0.875 | | 8 | 0.707 | 0.834 | | 9 | 0.666 | 0.798 | | 10 | 0.632 | 0.765 | | 11 | 0.602 | 0.735 | | 12 | 0.576 | 0.708 | | 13 | 0.553 | 0.684 | | 14 | 0.532 | 0.661 | | 15 | 0.514 | 0.641 | | 16 | 0.497 | 0.623 | | 17 | 0.482 | 0.606 | | 18 | 0.468 | 0.590 | | 19 | 0.456 | 0.575 | | 20 | 0.444 | 0.561 | | 25 | 0.396 | 0.505 | | 30 | 0.361 | 0.463 | | 35 | 0.335 | 0.430 | | 40 | 0.312 | 0.402 | | 45 | 0.294
**Title: Evaluating Linear Correlation Between Weights of Discarded Paper and Glass**

**Introduction:**

A data set includes weights of garbage discarded in one week from 62 different households. The paired weights of paper and glass were used to obtain the results shown. Is there sufficient evidence to support the claim that there is a linear correlation between weights of discarded paper and glass? Use a significance level of α = 0.05.

**Data Table:**

- **Correlation matrix:**

  | Variables | Paper  | Glass   |
  |-----------|--------|---------|
  | Paper     | 1      | 0.1666  |
  | Glass     | 0.1666 | 1       |

**Hypotheses:**

- **Null Hypothesis (H₀):** ρ = 0
- **Alternative Hypothesis (H₁):** ρ ≠ 0

**Statistical Analysis:**

1. **Identify the test statistic, r:**

   - r = [Input field]
   - (Round to three decimal places as needed.)

2. **Identify the critical value(s):**

   - (Round to three decimal places as needed.)
   - Options:
     - A. There is one critical value at r = [Input field]
     - B. There are two critical values at r = ±[Input field]

**Conclusion:**

State the conclusion based on the test statistic and critical value:

- Because the absolute value of the test statistic is [Input field] the positive critical value, there [Dropdown: "is"/"is not"] sufficient evidence to support the claim that there is a linear correlation between the weights of discarded paper and glass for a significance level of α = 0.05.

This exercise helps understand the process of testing for linear correlation between two variables using correlation coefficients and critical values.
Transcribed Image Text:**Title: Evaluating Linear Correlation Between Weights of Discarded Paper and Glass** **Introduction:** A data set includes weights of garbage discarded in one week from 62 different households. The paired weights of paper and glass were used to obtain the results shown. Is there sufficient evidence to support the claim that there is a linear correlation between weights of discarded paper and glass? Use a significance level of α = 0.05. **Data Table:** - **Correlation matrix:** | Variables | Paper | Glass | |-----------|--------|---------| | Paper | 1 | 0.1666 | | Glass | 0.1666 | 1 | **Hypotheses:** - **Null Hypothesis (H₀):** ρ = 0 - **Alternative Hypothesis (H₁):** ρ ≠ 0 **Statistical Analysis:** 1. **Identify the test statistic, r:** - r = [Input field] - (Round to three decimal places as needed.) 2. **Identify the critical value(s):** - (Round to three decimal places as needed.) - Options: - A. There is one critical value at r = [Input field] - B. There are two critical values at r = ±[Input field] **Conclusion:** State the conclusion based on the test statistic and critical value: - Because the absolute value of the test statistic is [Input field] the positive critical value, there [Dropdown: "is"/"is not"] sufficient evidence to support the claim that there is a linear correlation between the weights of discarded paper and glass for a significance level of α = 0.05. This exercise helps understand the process of testing for linear correlation between two variables using correlation coefficients and critical values.
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