1. A regression was done for 20 cities with the latitude (Latitude), a measurement of the distance of a location on the Earth from the equator in degrees and the average January temperatures (Temp) as the dependent variable measure in °F. The regression equation is Temp = 48.4 – 0.31 × Latitude a) What is the value of the slope? What does it mean in the context of the problem? %3D b) If the coefficient of determination, R² = 66.2%, what does it mean in the context of the problem? %3D c) Calculate the temperature for a latitude of 45 degrees.
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.
![### Regression Analysis of Average January Temperatures by Latitude
1. A regression was done for 20 cities with the latitude (\( \textit{Latitude} \)), a measurement of the distance of a location on the Earth from the equator in degrees and the average January temperatures (\( \textit{Temp} \)) as the dependent variable measured in °F. The regression equation is:
\[ \textit{Temp} = 48.4 - 0.31 \times \textit{Latitude} \]
#### a) What is the value of the slope? What does it mean in the context of the problem?
**Answer:**
The value of the slope is -0.31. This means that for every degree increase in latitude (moving further from the equator), the average January temperature decreases by 0.31°F.
#### b) If the coefficient of determination, \( R^2 = 66.2\% \), what does it mean in the context of the problem?
**Answer:**
The coefficient of determination, \( R^2 = 66.2\% \), indicates that 66.2% of the variation in the average January temperatures can be explained by the variation in latitudes of the cities. This suggests a decent but not perfect fit for the regression model.
#### c) Calculate the temperature for a latitude of 45 degrees.
**Answer:**
Using the given regression equation:
\[ \textit{Temp} = 48.4 - 0.31 \times 45 \]
\[ \textit{Temp} = 48.4 - 13.95 \]
\[ \textit{Temp} = 34.45°F \]
Therefore, the predicted average January temperature for a city at a latitude of 45 degrees is 34.45°F.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fedb1c45e-845e-4ed7-afc8-deed75acb3e2%2F2c8b4dac-bb9a-46e5-80f1-c04b5ed809a0%2Fpw8awi.jpeg&w=3840&q=75)
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