a particular professor has noticed that the number of people, p, who complain about his attitude is dependent on the number of cups of coffee, n, he drinks. from eight days of tracking he compiled the following data:

| people (p) | 12 | 10 | 10 | 6 | 7 | 5 | 3 | 4 |
| cups of coffee (n) | 1 | 1 | 2 | 3 | 3 | 4 | 5 | 5 |

unless otherwise stated, you can round values to two decimal places.

a) using regression to find a linear equation for ( p(n) )
( p(n) = )
b) interpret the meaning of the slope of your formula in the context of the problem. this means you need to use the slope to explain the relationship between cups of coffee the professor drinks and the number of people who complain about his attitude.
c) interpret the meaning of the ( p ) intercept in the context of the problem. think about whether ( p ) represents the ( x ) value or the ( y ) value to tell whether it is cups of coffee or people complaining.
d) use your model to predict the number of people that will complain about his attitude if he drinks 10 cups of coffee.
e) is the answer to part d reasonable? why or why not?
f) how many cups of coffee should he drink so that no one will complain about his attitude? round to more than one decimal place. think about whether cups of coffee is the ( x ) or ( y ) of the model, so you know where to plug in the 0.

Question

a particular professor has noticed that the number of people, p, who complain about his attitude is dependent on the number of cups of coffee, n, he drinks. from eight days of tracking he compiled the following data:

| people (p) | 12 | 10 | 10 | 6 | 7 | 5 | 3 | 4 |
| cups of coffee (n) | 1 | 1 | 2 | 3 | 3 | 4 | 5 | 5 |

unless otherwise stated, you can round values to two decimal places.

a) using regression to find a linear equation for ( p(n) )
( p(n) = )
b) interpret the meaning of the slope of your formula in the context of the problem. this means you need to use the slope to explain the relationship between cups of coffee the professor drinks and the number of people who complain about his attitude.
c) interpret the meaning of the ( p ) intercept in the context of the problem. think about whether ( p ) represents the ( x ) value or the ( y ) value to tell whether it is cups of coffee or people complaining.
d) use your model to predict the number of people that will complain about his attitude if he drinks 10 cups of coffee.
e) is the answer to part d reasonable? why or why not?
f) how many cups of coffee should he drink so that no one will complain about his attitude? round to more than one decimal place. think about whether cups of coffee is the ( x ) or ( y ) of the model, so you know where to plug in the 0.

Accepted Answer

### Part (a)
# Explanation:
## Step 1: Calculate necessary sums
We have the data points:
- $ n $: 1, 1, 2, 3, 3, 4, 5, 5
- $ P $: 12, 10, 10, 6, 7, 5, 3, 4

First, calculate $ \sum n $, $ \sum P $, $ \sum nP $, and $ \sum n^2 $.

$ \sum n = 1 + 1 + 2 + 3 + 3 + 4 + 5 + 5 = 24 $

$ \sum P = 12 + 10 + 10 + 6 + 7 + 5 + 3 + 4 = 57 $

$ \sum nP = (1\times12) + (1\times10) + (2\times10) + (3\times6) + (3\times7) + (4\times5) + (5\times3) + (5\times4) $
$ = 12 + 10 + 20 + 18 + 21 + 20 + 15 + 20 = 136 $

$ \sum n^2 = (1^2) + (1^2) + (2^2) + (3^2) + (3^2) + (4^2) + (5^2) + (5^2) $
$ = 1 + 1 + 4 + 9 + 9 + 16 + 25 + 25 = 90 $

## Step 2: Calculate the slope ($ m $) and intercept ($ b $) for linear regression
The formula for the slope $ m $ of the linear regression line $ P(n) = mn + b $ is:

$ m = \frac{n\sum nP - \sum n \sum P}{n\sum n^2 - (\sum n)^2} $

where $ n = 8 $ (number of data points).

Substitute the values:

$ m = \frac{8\times136 - 24\times57}{8\times90 - 24^2} $

First, calculate the numerator: $ 8\times136 = 1088 $, $ 24\times57 = 1368 $, so numerator $ = 1088 - 1368 = -280 $

Denominator: $ 8\times90 = 720 $, $ 24^2 = 576 $, so denominator $ = 720 - 576 = 144 $

Thus, $ m = \frac{-280}{144} \approx -1.94 $

The formula for the intercept $ b $ is:

$ b = \frac{\sum P - m\sum n}{n} $

Substitute the values:

$ b = \frac{57 - (-1.94)\times24}{8} $

Calculate $ (-1.94)\times24 \approx -46.56 $, so $ 57 - (-46.56) = 57 + 46.56 = 103.56 $

Then, $ b = \frac{103.56}{8} \approx 12.95 $

So the linear equation is $ P(n) = -1.94n + 12.95 $

# Answer:
$ P(n) = -1.94n + 12.95 $

### Part (b)
# Brief Explanations:
The slope of the linear equation $ P(n) = mn + b $ is $ m \approx -1.94 $. In the context of the problem, the slope represents the change in the number of people who complain ($ P $) for a one - unit increase in the number of cups of coffee ($ n $) the professor drinks. A negative slope means that for each additional cup of coffee the professor drinks, the number of people who complain about his attitude decreases by approximately 1.94 on average.

# Answer:
For each additional cup of coffee the professor drinks, the number of people who complain about his attitude decreases by approximately 1.94 on average.

### Part (c)
# Brief Explanations:
The $ P $-intercept (when $ n = 0 $) of the linear equation $ P(n)=- 1.94n + 12.95 $ is $ b\approx12.95 $. In the context of the problem, $ n $ is the number of cups of coffee and $ P $ is the number of people who complain. The $ P $-intercept represents the number of people who would complain about the professor's attitude if he drinks 0 cups of coffee. However, we need to consider if this is a meaningful interpretation in the real - world context. Since the professor can't drink a negative number of cups of coffee, the value at $ n = 0 $ is a theoretical value that may not be completely realistic, but it is the predicted number of complainers when no coffee is consumed.

# Answer:
The $ P $-intercept ($ \approx12.95 $) represents the predicted number of people who would complain about the professor's attitude if he drinks 0 cups of coffee. It is a theoretical value as drinking 0 cups of coffee is a boundary case.

### Part (d)
# Explanation:
## Step 1: Substitute $ n = 10 $ into the linear equation
We have the linear equation $ P(n)=-1.94n + 12.95 $. Substitute $ n = 10 $ into the equation:

$ P(10)=-1.94\times10 + 12.95 $

## Step 2: Calculate the value
First, calculate $ -1.94\times10=-19.4 $

Then, $ P(10)=-19.4 + 12.95=-6.45 $

But the number of people can't be negative. However, based on the model, the calculation is as above.

# Answer:
$ - 6.45 $ (Note: The negative value is not realistic in terms of the number of people, but it is the result from the linear model.)

### Part (e)
# Brief Explanations:
The answer to part (d) is not reasonable. The number of people who complain about the professor's attitude cannot be negative. The linear model is an approximation of the relationship between the number of cups of coffee and the number of complainers. When we extrapolate the model to $ n = 10 $ (which is far outside the range of the original data $ n = 1$ to $ n = 5$), the model gives a non - realistic negative value for the number of people.

# Answer:
The answer is not reasonable. The number of people can't be negative, and extrapolating the model to $ n = 10 $ (outside the data range) gives a non - realistic result.

### Part (f)
# Explanation:
## Step 1: Set $ P(n)=0 $ and solve for $ n $
We have the equation $ 0=-1.94n + 12.95 $

## Step 2: Solve for $ n $
First, rearrange the equation to isolate $ n $:

$ 1.94n = 12.95 $

Then, $ n=\frac{12.95}{1.94}\approx6.67 $

# Answer:
$ \approx6.67 $ cups of coffee.

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QUESTION IMAGE

Question

Explanation:

Part (a)

Step 1: Calculate necessary sums

Step 2: Calculate the slope (\( m \)) and intercept (\( b \)) for linear regression

Answer:

Part (b)