Questions: 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 α=0.05. Click here to view a table of critical values for the correlation coefficient Correlation matrix: Variables Paper Glass Paper 1 0.3543 Glass 0.3543 1 Determine the null and alternative hypotheses. H0: ρ H1: ρ (Type integers or decimals. Do not round.) Identify the correlation coefficient, r. r= (Round to three decimal places as needed.) Identify the critical value(s). (Round to three decimal places as needed.) A. There is one critical value at r= . B. There are two critical values at r= ±

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 α=0.05. Click here to view a table of critical values for the correlation coefficient Correlation matrix: Variables Paper Glass Paper 1 0.3543 Glass 0.3543 1 Determine the null and alternative hypotheses. H0: ρ H1: ρ (Type integers or decimals. Do not round.) Identify the correlation coefficient, r. r= (Round to three decimal places as needed.) Identify the critical value(s). (Round to three decimal places as needed.) A. There is one critical value at r= . B. There are two critical values at r= ±
Transcript text: 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 $\alpha=0.05$. Click here to view a table of critical values tor the correlation coeflicient Correlation matrix: \begin{tabular}{|l|r|r|} \hline Variables & Paper & Glass \\ \hline Paper & 1 & 0.3543 \\ \hline Glass & 0.3543 & 1 \\ \hline \end{tabular} Determine the null and alternative hypotheses. $H_{0}: \rho$ $\square$ $\square$ $H_{1}: \rho$ $\square$ $\square$ (Type integers or decimals. Do not round.) Identify the correlation coefficient, r. $\mathrm{r}=$ $\square$ (Round to three decimal places as needed.) Identify the critical value(s). (Round to three decimal places as needed.) A. There is one critical value at $r=$ $\square$ . B. There are two critical values at $r= \pm$ $\square$
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Solution

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Solution Steps

Step 1: Hypotheses

We set up the null and alternative hypotheses as follows:

  • \( H_0: \rho = 0 \) (There is no linear correlation between weights of discarded paper and glass)
  • \( H_1: \rho \neq 0 \) (There is a linear correlation between weights of discarded paper and glass)
Step 2: Correlation Coefficient

The calculated correlation coefficient \( r \) is: \[ r = 0.354 \]

Step 3: Critical Values

For a significance level of \( \alpha = 0.05 \) and a sample size of \( n = 62 \), the critical values for the correlation coefficient are: \[ r = \pm 0.250 \]

Step 4: Decision

To determine whether to reject the null hypothesis, we compare the absolute value of the correlation coefficient with the critical value: \[ |r| = 0.354 > 0.250 \] Since \( |r| \) exceeds the critical value, we reject the null hypothesis.

Final Answer

There is sufficient evidence to support the claim of a linear correlation between weights of discarded paper and glass.

\(\boxed{H_0 \text{ is rejected}}\)

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