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Amélie runs a bakery. She wants to find out whether her cake sales are affected by the weather conditions. She recorded the daily temperature and the number of cakes she sold on different days of the year. The table shows the data she gathered.




Daily Temperature (°F)		Cakes Sold

42		39

45		52

48		31

54		61

59		72

62		35

64		61

65		34

67		58

75		45

84		24






The correlation coefficient is close to [Choose an answer].

Based on this information, we can conclude that Amélie's cake sales are [Choose an answer] affected by the daily temperature.

Verified Answer :

  • The calculated Pearson correlation coefficient is \(r \approx -0.20\).
  • A coefficient near \(0\) indicates a weak or no linear relationship.
  • Consequently, we conclude that the cake sales are not significantly affected by the daily temperature.
  • Final answer: The correlation coefficient is close to \(\boxed{0}\) and the sales are \(\boxed{\text{not significantly}}\) affected by the temperature.

Explanation

  1. Calculate the means of the data sets

To determine the relationship between temperature and cake sales, we calculate the Pearson correlation coefficient \(r\). The coefficient \(r\) measures the strength and direction of a linear relationship between two variables, ranging from \(-1\) to \(1\).
First, we find the mean temperature \(\bar{x}\) and the mean cake sales \(\bar{y}\):
\[\bar{x} = \frac{42+45+48+54+59+62+64+65+67+75+84}{11} \approx 60.45\]
\[\bar{y} = \frac{39+52+31+61+72+35+61+34+58+45+24}{11} \approx 46.55\]

  1. Compute the correlation coefficient r

Next, we use the formula for the Pearson correlation coefficient:
\[r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}\]
By calculating the deviations from the means for each pair, the sum of the products of deviations \(\sum (x_i - \bar{x})(y_i - \bar{y})\) is computed. Then, the sum of the squared deviations for both variables \(\sum (x_i - \bar{x})^2\) and \(\sum (y_i - \bar{y})^2\) are calculated to find the denominator.
Substituting these values, we find that the correlation coefficient is approximately \(r \approx -0.20\).

  1. Analyze the result and draw a conclusion

A correlation coefficient \(r\) close to \(0\) (such as \(-0.20\)) indicates a very weak or negligible linear relationship between the two variables. This suggests that as the temperature changes, there is no consistent linear trend in the number of cakes sold. Therefore, we can conclude that Amélie's cake sales are not significantly affected by the daily temperature.

Examples

In real life, correlation analysis helps businesses understand if external factors like weather, holiday seasons, or marketing campaigns drive sales. If a bakery notices a correlation of \(r \approx 0.9\) between temperature and ice cream sales, they would know to increase production on hot days. If the correlation is near \(0\), they look for other variables, such as day of the week or local events, to explain sales trends. Understanding these relationships allows owners to make data-driven decisions for inventory and staffing.

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