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?

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?

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 $) of the linear regression line $ P(n) = mn + b $
The formula for the slope $ m $ 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} $
$ = \frac{1088 - 1368}{720 - 576} $
$ = \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} $
$ = \frac{57 + 46.56}{8} $
$ = \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 $. In the context of this problem, the slope represents the change in the number of people who complain ($ 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 people.

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

### 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, the $ P $-intercept represents the number of people who would complain about the professor's attitude when he drinks 0 cups of coffee. Since $ P $ is the number of people complaining, the $ P $-intercept (when $ n = 0 $) gives an estimate of how many people would complain if the professor had no coffee.

# Answer:
The $ P $-intercept (approximately 12.95) represents the estimated number of people who would complain about the professor's attitude when he drinks 0 cups of coffee.

### Part (d)
# Explanation:
## Step 1: Use the linear model
We have the linear model $ P(n)=-1.94n + 12.95 $. We want to find $ P $ when $ n = 10 $.

Substitute $ n = 10 $ into the equation:

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

## Step 2: Calculate the value
$ P(10)=-19.4+12.95=-6.45 $

# Answer:
$ - 6.45 $ (Note: A negative number of people complaining may not be practically meaningful, but this is the result from the model)

### Part (e)
# Brief Explanations:
The answer to part (d) is not reasonable. The number of people who can complain about the professor's attitude cannot be negative. Also, the value of $ n = 10 $ cups of coffee is far outside the range of the original data (the original $ n $ values range from 1 to 5). Extrapolating the linear model far beyond the range of the original data can lead to unreliable and in this case, non - sensical (negative number of people) results.

# Answer:
The answer to part (d) is not reasonable. The number of people complaining cannot be negative, and $ n = 10 $ is far outside the original data range, leading to an unreliable prediction.

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QUESTION IMAGE

Question

Explanation:

Part (a)

Step 1: Calculate necessary sums

Step 2: Calculate the slope (\( m \)) and intercept (\( b \)) of the linear regression line \( P(n) = mn + b \)

Answer:

Part (b)