Questions: A calculator is allowed on this question. The 12th grade class is having a cookie sale every day after school for one week. They decide to offer samples of cookies to encourage customers to buy. The table below shows the number of samples that were given away and the amount of money they made in sales for each day. Approximately how much should the students expect to make if they hold a cookie sale for one more day and give away 25 samples? Number of Samples (x) Sales (f(x)) 12 30.25 20 52.50 18 50.00 13 38.75 16 42.00

A calculator is allowed on this question. The 12th grade class is having a cookie sale every day after school for one week. They decide to offer samples of cookies to encourage customers to buy. The table below shows the number of samples that were given away and the amount of money they made in sales for each day. Approximately how much should the students expect to make if they hold a cookie sale for one more day and give away 25 samples? Number of Samples (x) Sales (f(x)) 12 30.25 20 52.50 18 50.00 13 38.75 16 42.00
Transcript text: A calculator is allowed on this question. The 12 th grade class is having a cookie sale every day after school for one week. They decide to offer samples of cookies to encourage customers to buy. The table below shows the number of samples that were given away and the amount of money they made in sales for each day. Approximately how much should the students expect to make if they hold a cookie sale for one more day and give away 25 samples? Number of Samples $(x)$ & Sales $(f(x))$ \\ 12 & $\$ 30.25$ \\ 20 & $\$ 52.50$ \\ 18 & $\$ 50.00$ \\ 13 & $\$ 38.75$ \\ 16 & $\$ 42.00$
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Solution

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Solution Steps

To estimate the sales for 25 samples, we can use linear regression to find the best-fit line for the given data points. This line will help us predict the sales for a given number of samples. We will use the least squares method to calculate the slope and intercept of the line, and then use the equation of the line to predict the sales for 25 samples.

Step 1: Organize the Data

We have the following data points for the number of samples given away and the corresponding sales:

  • \( (12, 30.25) \)
  • \( (20, 52.50) \)
  • \( (18, 50.00) \)
  • \( (13, 38.75) \)
  • \( (16, 42.00) \)
Step 2: Determine the Linear Regression Line

To find the best-fit line, we use the linear regression formula \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. From the calculations, we have:

  • Slope (\( m \)) = 2.5770
  • Intercept (\( b \)) = 1.9833

Thus, the equation of the line is: \[ f(x) = 2.5770x + 1.9833 \]

Step 3: Predict Sales for 25 Samples

To predict the sales for 25 samples, substitute \( x = 25 \) into the equation: \[ f(25) = 2.5770 \times 25 + 1.9833 \]

Calculating this gives: \[ f(25) = 64.425 + 1.9833 = 66.4083 \]

Final Answer

\(\boxed{66.41}\)

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