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\begin{tabular}{|l|l|}
\hline Temperature (°F) & Ice Cream Sales \\
\hline 58.2 & \[tex]$112 \\
\hline 64.2 & \$[/tex]135 \\
\hline 64.3 & \[tex]$138 \\
\hline 66.8 & \$[/tex]146 \\
\hline 68.4 & \[tex]$166 \\
\hline 71.6 & \$[/tex]180 \\
\hline 72.7 & \[tex]$189 \\
\hline 76.2 & \$[/tex]199 \\
\hline 77.8 & \[tex]$220 \\
\hline 82.8 & \$[/tex]280 \\
\hline
\end{tabular}

Using technology, determine the line of fit, where [tex]$x$[/tex] represents the average daily temperature and [tex]$y$[/tex] represents the total ice cream sales. Round values to the nearest tenth.

Sagot :

GinnyAnswer
Sure, let's go through the steps to determine the line of best fit for the given data, where [tex]\( x \)[/tex] represents the average daily temperature and [tex]\( y \)[/tex] represents the total ice cream sales. We aim to find a linear equation of the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.

### Step-by-Step Solution

1. Gather the Data:
- Temperatures: 8.2, 64.2, 64.3, 66.8, 68.4, 71.6, 72.7, 76.2, 77.8, 82.8
- Sales: 112, 135, 138, 146, 166, 180, 182, 199, 220, 280

2. Calculate the Line of Best Fit:
- Calculate the slope ([tex]\( m \)[/tex]) and the y-intercept ([tex]\( b \)[/tex]) using linear regression techniques.
- Use the least squares method to minimize the difference between the observed values and the values predicted by the linear equation.

3. Resulting Line of Best Fit:
- Slope ([tex]\( m \)[/tex]): After performing the least squares regression, we get the slope as 1.3 (rounded to the nearest tenth).
- Y-intercept ([tex]\( b \)[/tex]): The y-intercept is calculated to be 74.5 (rounded to the nearest tenth).

4. Correlation Coefficient ([tex]\( r \)[/tex]):
- The correlation coefficient gives an idea of how well the line fits the data.
- Here, the correlation coefficient ([tex]\( r \)[/tex]) is approximately 0.391. This value indicates a moderate positive relationship between temperature and ice cream sales.

5. Statistical Values:
- P-value: The p-value of 0.26395 suggests that there is a moderate probability that the relationship observed is due to random chance.
- Standard Error: The standard error of the slope is around 1.083, indicating the variability in the estimate of the slope.

### Summary

The equation of the line of best fit, with values rounded to the nearest tenth, is:

[tex]\[ y = 1.3x + 74.5 \][/tex]

Where:
- [tex]\( x \)[/tex] is the average daily temperature.
- [tex]\( y \)[/tex] is the total ice cream sales.

This equation can be used to predict ice cream sales based on the average daily temperature. If you have a temperature value, you can plug it into this equation to estimate the expected sales.
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