To analyze the given scenarios based on the trend line provided in the scatterplot, we first need to determine the relationship between the hours open and the number of customers.
From the trend line showing points (1, 25) and (2, 50), we can deduce the equation of the trend line. This line indicates that for every hour the store is open, the number of customers increases by about 25. This suggests a linear relationship.
To write the equation of the trend line: The slope (m) can be calculated from the two points: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{50 - 25}{2 - 1} = 25. \]
The y-intercept (b) can be inferred from the point (1, 25): If 1 hour corresponds to 25 customers, when we use the point-slope form: \[ y = mx + b \Rightarrow 25 = 25(1) + b \Rightarrow b = 0. \]
Thus, the equation of the trend line is: \[ y = 25x, \] where \(y\) is the total number of customers and \(x\) is the number of hours open.
Now, let’s calculate the expected number of customers for 7 hours: \[ y = 25(7) = 175. \]
Now we will analyze each scenario:
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After 1/2 hour, 1 customer had been in the store.
- Expected: \( 25 \times 0.5 = 12.5 \) customers. So, this scenario is unlikely.
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After 6 hours, 129 customers had been in the store.
- Expected: \( 25 \times 6 = 150 \) customers. This scenario is unlikely because it’s significantly under the expected value.
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After 7 hours, 175 customers had been in the store.
- Expected: \( 25 \times 7 = 175 \) customers. This aligns perfectly with the trend line.
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After 8.5 hours, 225 customers had been in the store.
- Expected: \( 25 \times 8.5 = 212.5 \) customers. This is higher than what we would expect.
Based on this analysis, the most likely scenario is: After 7 hours, 175 customers had been in the store.
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