Many attempts have been made to relate happiness with various factors. One such study relates happiness with age and finds that holding everything else constant, people are least happy when they are in their mid-40s (The Economist, December 16, 2010). The accompanying table shows a portion of data on a respondent’s age and his/her perception of well-being on a scale from 0 to 100. [You may find it useful to reference the t table.] Happiness Age 62 49 66 51 ⋮ ⋮ 72 69 Click here for the Excel Data File a-1. Calculate the sample correlation coefficient between age and happiness. (Round intermediate calculations to at least 4 decimal places and final answer to 4 decimal places.) a-2. Interpret the sample correlation coefficient between age and happiness. multiple choice 1 The correlation coefficient indicates a positive linear relationship. The correlation coefficient indicates a negative linear relationship. The correlation coefficient indicates no linear relationship. b-1. Specify the competing hypotheses in order to determine whether the population correlation between the age and happiness differs from zero. multiple choice 2 H0: ρxy = 0; HA: ρxy ≠ 0 H0: ρxy ≤ 0; HA: ρxy > 0 H0: ρxy ≥ 0; HA: ρxy < 0 b-2. Calculate the value of the test statistic. (Round intermediate calculations to at least 4 decimal places and final answer to 3 decimal places.) b-3. Find the p-value. multiple choice 3 p-value < 0.01 0.01 ≤ p-value < 0.02 0.02 ≤ p-value < 0.05 0.05 ≤ p-value < 0.10 p-value ≥ 0.10 b-4. Is the correlation coefficient statistically significant at the 1% level? multiple choice 4 Yes, since we reject H0. Yes, since we do not reject H0. No, since we reject H0. No, since we do not reject H0. c. Use a scatterplot to determine if there is a flaw with the above correlation analysis. multiple choice 5 Flawed since the scatterplot shows a linear relationship. Flawed since the scatterplot shows a nonlinear relationship. Not flawed since the scatterplot shows a linear relationship. Not flawed since the scatterplot shows a nonlinear relationship.
Correlation
Correlation defines a relationship between two independent variables. It tells the degree to which variables move in relation to each other. When two sets of data are related to each other, there is a correlation between them.
Linear Correlation
A correlation is used to determine the relationships between numerical and categorical variables. In other words, it is an indicator of how things are connected to one another. The correlation analysis is the study of how variables are related.
Regression Analysis
Regression analysis is a statistical method in which it estimates the relationship between a dependent variable and one or more independent variable. In simple terms dependent variable is called as outcome variable and independent variable is called as predictors. Regression analysis is one of the methods to find the trends in data. The independent variable used in Regression analysis is named Predictor variable. It offers data of an associated dependent variable regarding a particular outcome.
Many attempts have been made to relate happiness with various factors. One such study relates happiness with age and finds that holding everything else constant, people are least happy when they are in their mid-40s (The Economist, December 16, 2010). The accompanying table shows a portion of data on a respondent’s age and his/her perception of well-being on a scale from 0 to 100. [You may find it useful to reference the t table.]
| Happiness | Age | ||||
| 62 | 49 | ||||
| 66 | 51 | ||||
| ⋮ | ⋮ | ||||
| 72 | 69 | ||||
Click here for the Excel Data File
a-1. Calculate the sample
a-2. Interpret the sample correlation coefficient between age and happiness.
multiple choice 1
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The correlation coefficient indicates a positive linear relationship.
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The correlation coefficient indicates a negative linear relationship.
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The correlation coefficient indicates no linear relationship.
b-1. Specify the competing hypotheses in order to determine whether the population correlation between the age and happiness differs from zero.
multiple choice 2
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H0: ρxy = 0; HA: ρxy ≠ 0
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H0: ρxy ≤ 0; HA: ρxy > 0
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H0: ρxy ≥ 0; HA: ρxy < 0
b-2. Calculate the value of the test statistic. (Round intermediate calculations to at least 4 decimal places and final answer to 3 decimal places.)
b-3. Find the p-value.
multiple choice 3
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p-value < 0.01
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0.01 ≤ p-value < 0.02
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0.02 ≤ p-value < 0.05
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0.05 ≤ p-value < 0.10
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p-value ≥ 0.10
b-4. Is the correlation coefficient statistically significant at the 1% level?
multiple choice 4
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Yes, since we reject H0.
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Yes, since we do not reject H0.
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No, since we reject H0.
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No, since we do not reject H0.
c. Use a scatterplot to determine if there is a flaw with the above correlation analysis.
multiple choice 5
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Flawed since the scatterplot shows a linear relationship.
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Flawed since the scatterplot shows a nonlinear relationship.
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Not flawed since the scatterplot shows a linear relationship.
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Not flawed since the scatterplot shows a nonlinear relationship.
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