Ash 2 80

Question

Ben 1 30
Carl 4 175
Dave 2 125
Ed 3 140
Jill 4 150
Mandy 2.5 100
Pam 3 100
Rachel 3.5 225
Sue 1.5 50

Draw a trend line for your scatter plot.

Determine the correlation present in your scatter plot.

Answer 1

To draw a trend line for a scatter plot based on the data you've provided, we first need to identify the two variables. Based on your data, it seems like the first number represents one variable (let's call it "X") and the second number represents another variable (let's call it "Y").

Here is your data represented as pairs of (X, Y):

Ash   (2, 80)
Ben   (1, 30)
Carl  (4, 175)
Dave  (2, 125)
Ed    (3, 140)
Jill  (4, 150)
Mandy (2.5, 100)
Pam   (3, 100)
Rachel(3.5, 225)
Sue   (1.5, 50)

Plotting the Data

To draw a scatter plot:

1. Place "X" values on the horizontal axis and "Y" values on the vertical axis.
2. Plot each point based on its (X, Y) coordinates.

Drawing a Trend Line

A trend line is often estimated via linear regression. However, since we don't have graphing tools here, I'll describe how you can calculate it conceptually:

1. Calculate the Mean of X and Y:

   • Mean of X: (2 + 1 + 4 + 2 + 3 + 4 + 2.5 + 3 + 3.5 + 1.5) / 10 = 2.5
   • Mean of Y: (80 + 30 + 175 + 125 + 140 + 150 + 100 + 100 + 225 + 50) / 10 = 1150 / 10 = 115

2. Use the Least Squares Method to find the slope (m) and y-intercept (b):

   • Slope (m) = (Σ((X - Mean_X)(Y - Mean_Y))) / (Σ((X - Mean_X)²))
   • Y-intercept (b) = Mean_Y - m * Mean_X

3. You can estimate the linear relationship from your calculated parameters: \[ Y = mX + b \]

Determine the Correlation

To determine the correlation coefficient (r), you can use the formula: \[ r = \frac{n(ΣXY) - ΣXΣY}{\sqrt{[nΣX² - (ΣX)²][nΣY² - (ΣY)²]}} \] Where:

• n is the number of pairs,
• ΣXY is the sum of the product of each pair,
• ΣX is the sum of all X values,
• ΣY is the sum of all Y values.

Alternatively, you can visually inspect the scatter plot:

• If the points tend to rise together, we have a positive correlation.
• If one goes down as the other goes up, we have a negative correlation.
• The closer the points are to a straight line, the stronger the correlation.

Summary

1. Plot the points (scatter plot).
2. Calculate the trend line using the method of least squares.
3. Determine the correlation coefficient using the formula provided.

You can use statistical software or a graphing calculator to perform these calculations and create the scatter plot to visualize the trend line accurately. If you did it manually or with basic tools, you would get an approximate trend line and correlation.

Answer 2

Use information from number 3 to answer the following question:
How many words can a student type in 8 minutes?

How many words can a student type in 15 minutes?

Approximately how many minutes will it take a student to type a 2,500 word essay.

Answer 3

To answer the questions about typing speed based on the data you provided, we need to establish a relation between the time taken (in minutes) and the number of words typed (based on the provided data points).

Step 1: Analyze the Data

We can summarize the data in terms of typing speed (words per minute) from the provided data points (let's denote the first number as the time in minutes and the second as the number of words typed):

• Ash: 2 min, 80 words → 40 words/min
• Ben: 1 min, 30 words → 30 words/min
• Carl: 4 min, 175 words → 43.75 words/min
• Dave: 2 min, 125 words → 62.5 words/min
• Ed: 3 min, 140 words → 46.67 words/min
• Jill: 4 min, 150 words → 37.5 words/min
• Mandy: 2.5 min, 100 words → 40 words/min
• Pam: 3 min, 100 words → 33.33 words/min
• Rachel: 3.5 min, 225 words → 64.29 words/min
• Sue: 1.5 min, 50 words → 33.33 words/min

Step 2: Calculate Average Typing Speed

To simplify calculations, let’s find the average typing speed from the given data.

\[ \text{Average Speed} = \frac{\text{Sum of all words typed}}{\text{Sum of all minutes}} \]

Calculating total sums:

• Total Words Typed: 80 + 30 + 175 + 125 + 140 + 150 + 100 + 100 + 225 + 50 = 1,175 words
• Total Minutes: 2 + 1 + 4 + 2 + 3 + 4 + 2.5 + 3 + 3.5 + 1.5 = 23.5 minutes

Now calculate the average:

\[ \text{Average Speed} = \frac{1175}{23.5} \approx 50 words/min \]

Step 3: Answering the Questions

1. How many words can a student type in 8 minutes?

   Using the average speed: \[ \text{Words in 8 minutes} = \text{Average Speed} \times \text{Time} = 50 \text{ words/min} \times 8 \text{ min} = 400 \text{ words} \]

2. How many words can a student type in 15 minutes?

   Using the average speed: \[ \text{Words in 15 minutes} = 50 \text{ words/min} \times 15 \text{ min} = 750 \text{ words} \]

3. Approximately how many minutes will it take a student to type a 2,500-word essay?

   To find the time required: \[ \text{Time} = \frac{\text{Total Words}}{\text{Average Speed}} = \frac{2500 \text{ words}}{50 \text{ words/min}} = 50 \text{ minutes} \]

Summary of Answers

• A student can type approximately 400 words in 8 minutes.
• A student can type approximately 750 words in 15 minutes.
• It will take a student approximately 50 minutes to type a 2,500-word essay.

Answer 4

Determine the correlation present in your scatter plot.

Answer 5

To determine the correlation present in your scatter plot based on the data provided, we first need to calculate the correlation coefficient (often denoted as \( r \)). The correlation coefficient measures the strength and direction of a linear relationship between two variables.

Data Recap

Here are the data points we are working with:

\[ \begin{align*} \text{Ash} & : (2, 80) \ \text{Ben} & : (1, 30) \ \text{Carl} & : (4, 175) \ \text{Dave} & : (2, 125) \ \text{Ed} & : (3, 140) \ \text{Jill} & : (4, 150) \ \text{Mandy} & : (2.5, 100) \ \text{Pam} & : (3, 100) \ \text{Rachel} & : (3.5, 225) \ \text{Sue} & : (1.5, 50) \ \end{align*} \]

Step-by-Step Calculation of the Correlation Coefficient

1. Calculate necessary sums

Let’s define the variables:

• \( X \): time (minutes)
• \( Y \): words typed

Now calculate the sums we need:

• n (number of data points) = 10
• Sum of X: \( ΣX = 2 + 1 + 4 + 2 + 3 + 4 + 2.5 + 3 + 3.5 + 1.5 = 23.5 \)
• Sum of Y: \( ΣY = 80 + 30 + 175 + 125 + 140 + 150 + 100 + 100 + 225 + 50 = 1,175 \)
• Sum of XY: \[ ΣXY = (2 \cdot 80) + (1 \cdot 30) + (4 \cdot 175) + (2 \cdot 125) + (3 \cdot 140) + (4 \cdot 150) + (2.5 \cdot 100) + (3 \cdot 100) + (3.5 \cdot 225) + (1.5 \cdot 50) \] Breaking it down: \[ = 160 + 30 + 700 + 250 + 420 + 600 + 250 + 300 + 787.5 + 75 = 3,572.5 \]
• Sum of X²: \[ ΣX² = 2^2 + 1^2 + 4^2 + 2^2 + 3^2 + 4^2 + 2.5^2 + 3^2 + 3.5^2 + 1.5^2 \] Breaking it down: \[ = 4 + 1 + 16 + 4 + 9 + 16 + 6.25 + 9 + 12.25 + 2.25 = 80.5 \]
• Sum of Y²: \[ ΣY² = 80^2 + 30^2 + 175^2 + 125^2 + 140^2 + 150^2 + 100^2 + 100^2 + 225^2 + 50^2 \] Breaking it down: \[ = 6400 + 900 + 30625 + 15625 + 19600 + 22500 + 10000 + 10000 + 50625 + 2500 = 1,095,000 \]

2. Plugging it into the correlation formula

The formula for Pearson's correlation coefficient \( r \) is:

\[ r = \frac{n(ΣXY) - (ΣX)(ΣY)}{\sqrt{[nΣX² - (ΣX)²][nΣY² - (ΣY)²]}} \]

Using our calculations:

Substituting in:

• \( n = 10 \)
• \( ΣXY = 3572.5 \)
• \( ΣX = 23.5 \)
• \( ΣY = 1175 \)
• \( ΣX^2 = 80.5 \)
• \( ΣY^2 = 1095000 \)

Calculating the components: \[ 10(3572.5) - (23.5)(1175) = 35725 - 27612.5 = 8112.5 \] \[ \sqrt{[10(80.5) - (23.5)^2][10(1095000) - (1175)^2]} = \sqrt{[805 - 552.25][10950000 - 1380625]} = \sqrt{[252.75][9589375]} \]

Calculating the individual terms: \[ 252.75 \times 9589375 \approx 2420198437.5 \] Then take the square root: \[ \sqrt{2420198437.5} \approx 15563.14 \]

Now, combining this back into the correlation formula: \[ r \approx \frac{8112.5}{15563.14} \approx 0.52 \]

Interpretation

• The correlation coefficient \( r \approx 0.52 \) suggests a moderate positive correlation between the time spent typing and the number of words typed.

This indicates that, generally, as the number of minutes spent typing increases, the number of words typed also tends to increase. However, it is not a perfect correlation, suggesting some variability in the data.