To address this problem, we will proceed step by step:

1. Calculate the correlation coefficient between the number of shipments and delivery time.
2. Perform a simple linear regression to model the relationship between the number of shipments and delivery time.
3. Predict the delivery time for 30 shipments using the regression model.

Let me begin by solving each part mathematically.

Step 1: Correlation Coefficient Calculation

The formula for the correlation coefficient $r$ between two variables $X$ (number of shipments) and $Y$ (delivery time) is:

$r = \frac{n\sum{XY} - \sum{X}\sum{Y}}{\sqrt{\left(n\sum{X^2} - (\sum{X})^2\right)\left(n\sum{Y^2} - (\sum{Y})^2\right)}}$

Where:

• $X$ = Number of Shipments
• $Y$ = Delivery Time
• $n$ = Number of observations

The dataset provided is:

• Shipments ($X$): 55, 60, 40, 70, 55, 45, 65
• Delivery Time ($Y$): 4, 5, 5, 6, 6, 4, 5

Step 2: Simple Linear Regression

The simple linear regression equation is of the form: $\hat{Y} = \beta_0 + \beta_1 X$ Where:

• $\hat{Y}$ is the predicted delivery time.
• $X$ is the number of shipments.
• $\beta_0$ is the y-intercept.
• $\beta_1$ is the slope of the regression line.

To compute the slope ($\beta_1$) and intercept ($\beta_0$): $\beta_1 = \frac{n\sum{XY} - \sum{X}\sum{Y}}{n\sum{X^2} - (\sum{X})^2}$ $\beta_0 = \frac{\sum{Y} - \beta_1\sum{X}}{n}$

Step 3: Predict Delivery Time for 30 Shipments

Once we have the regression equation, we will substitute $X = 30$ to predict the delivery time.

I'll now calculate each step.Here are the results:

1. Correlation Coefficient: The correlation coefficient between the number of shipments and delivery time is approximately 0.482. This indicates a moderate positive relationship.

2. Linear Regression Equation: The equation modeling the relationship between the number of shipments ($X$) and delivery time ($Y$) is: $\hat{Y} = 2.93 + 0.037X$ where:

   • $\beta_0 = 2.93$ (intercept)
   • $\beta_1 = 0.037$ (slope)

3. Predicted Delivery Time for 30 Shipments: Using the regression model, if the number of shipments is 30, the predicted delivery time is approximately 4.04 days.

Would you like more details on any part of this process or have additional questions?

Here are five related questions:

1. What does a correlation coefficient of 0.482 imply in terms of the strength of the relationship?
2. How can we improve the accuracy of the prediction model?
3. What does the slope ($\beta_1$) of the regression equation tell us?
4. How does the predicted delivery time change as the number of shipments increases?
5. What are the assumptions underlying simple linear regression?

Tip: A higher correlation coefficient (closer to 1 or -1) indicates a stronger linear relationship between two variables.