An analysis conducted by the accounting department at Stan's Jeans found a linear relationship exists between the number of pairs a store will sell weekly and the price, in dollars, at which it sells them. The model for this relationship can be given by the equation pairs = -1.29 x (price) + 78.26 (a) Interpret the slope of the line in the context of the data.
An analysis conducted by the accounting department at Stan's Jeans found a linear relationship exists between the number of pairs a store will sell weekly and the price, in dollars, at which it sells them. The model for this relationship can be given by the equation pairs = -1.29 x (price) + 78.26 (a) Interpret the slope of the line in the context of the data.
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Author:Amos Gilat
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![### Linear Relationships in Sales: Jeans Sales and Pricing
An analysis conducted by the accounting department at Stan's Jeans found a linear relationship between the number of pairs of jeans a store will sell weekly and the price, in dollars, at which it sells them. The model for this relationship can be given by the equation:
\[ \widehat{\text{pairs}} = -1.29 \times (\text{price}) + 78.26 \]
#### (a) Interpreting the Slope
**Interpret the slope of the line in the context of the data:**
The slope of the line is \(-1.29\). In the context of the data, this suggests that for every one-dollar increase in the price of jeans, the store expects to sell approximately 1.29 fewer pairs of jeans weekly. This indicates an inverse relationship between price and the number of pairs sold.
#### (b) Calculating Residual Demand
**The residual demand (i.e. number of pairs) when a store sets the price at $37 is -2.53 pairs. How many pairs of jeans did the store sell at that price?**
To find the actual number of pairs sold when the price is $37, we start by substituting the price into the equation to find the predicted number of pairs.
\[ \widehat{\text{pairs}} = -1.29 \times 37 + 78.26 \]
Calculating this:
\[ \widehat{\text{pairs}} = -47.730 + 78.26 = 30.53 \]
The model predicts that the store will sell approximately 30.53 pairs of jeans at a price of $37. Given that the residual (the difference between the actual number sold and the number predicted by the model) is \(-2.53\), the actual number sold is:
\[ \text{Actual pairs} = \widehat{\text{pairs}} + \text{Residual} \]
\[ \text{Actual pairs} = 30.53 + (-2.53) \]
\[ \text{Actual pairs} = 28 \]
Therefore, the store sold 28 pairs of jeans at the price of $37.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F06a9a237-982c-465e-8a17-4be5fc663e3e%2F95c3ff0c-6e43-415c-846d-dacd141b464d%2F4lhfbg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Linear Relationships in Sales: Jeans Sales and Pricing
An analysis conducted by the accounting department at Stan's Jeans found a linear relationship between the number of pairs of jeans a store will sell weekly and the price, in dollars, at which it sells them. The model for this relationship can be given by the equation:
\[ \widehat{\text{pairs}} = -1.29 \times (\text{price}) + 78.26 \]
#### (a) Interpreting the Slope
**Interpret the slope of the line in the context of the data:**
The slope of the line is \(-1.29\). In the context of the data, this suggests that for every one-dollar increase in the price of jeans, the store expects to sell approximately 1.29 fewer pairs of jeans weekly. This indicates an inverse relationship between price and the number of pairs sold.
#### (b) Calculating Residual Demand
**The residual demand (i.e. number of pairs) when a store sets the price at $37 is -2.53 pairs. How many pairs of jeans did the store sell at that price?**
To find the actual number of pairs sold when the price is $37, we start by substituting the price into the equation to find the predicted number of pairs.
\[ \widehat{\text{pairs}} = -1.29 \times 37 + 78.26 \]
Calculating this:
\[ \widehat{\text{pairs}} = -47.730 + 78.26 = 30.53 \]
The model predicts that the store will sell approximately 30.53 pairs of jeans at a price of $37. Given that the residual (the difference between the actual number sold and the number predicted by the model) is \(-2.53\), the actual number sold is:
\[ \text{Actual pairs} = \widehat{\text{pairs}} + \text{Residual} \]
\[ \text{Actual pairs} = 30.53 + (-2.53) \]
\[ \text{Actual pairs} = 28 \]
Therefore, the store sold 28 pairs of jeans at the price of $37.
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