The president of a company that manufactures car seats has been concerned about the number and cost of machine breakdowns. The problem is that the machines are old and becoming quite unreliable. However, the cost of replacing them is quite high, and the president is not certain that the cost can be made up in today’s slow economy. To help make a decision about replacement, he gathered data about last month’s cost for repairs and the ages (in moths) of the plant’s 20 welding machines.

a. Find the sample regression.
b. Interpret the coefficients.
c. Determine the coefficient of determination, and discuss what this statistic tells you.
d. Conduct a test to determine whether the age of a machine and its monthly cost of repair are linearly related.
e. Is the fit of the simple linear model good enough to allow the president to predict the monthly repair cost of a welding machine that is 120 months old? If so, find a 95% prediction interval. If not, explain why not

To answer these questions, we need to perform a linear regression analysis on the data. Here are the steps to do so:

Step 1: Gather the data on the cost of repairs and ages of the plant's welding machines for last month.

Once you have the data, you can proceed with the analysis.

a. Find the sample regression:

To find the sample regression, we need to calculate the regression equation, which represents the relationship between the age of the machine and its monthly cost of repair.

The regression equation can be found using statistical software or Excel. It will have the form:

y = a + bx

where y represents the monthly cost of repair and x represents the age of the machine. The values of 'a' and 'b' will be calculated based on the data.

b. Interpret the coefficients:

The coefficient 'a' (intercept) represents the estimated monthly cost of repair when the machine is brand new (age = 0). Interpretation of 'a' can be something like "the estimated cost of repairs for a new machine is $X."

The coefficient 'b' (slope) represents the estimated change in the monthly cost of repair for each additional month of the machine's age. Interpretation of 'b' can be something like "for every additional month of age, the estimated monthly cost of repairs increases/decreases by $X."

c. Determine the coefficient of determination and discuss what this statistic tells you:

The coefficient of determination (R-squared) tells us the proportion of the variation in the monthly cost of repair that can be explained by the age of the machine. It ranges from 0 to 1, with 1 indicating a perfect fit.

To calculate R-squared, you need to square the correlation coefficient (r) derived from the regression analysis. R-squared can be interpreted as the percentage of the variation in the monthly cost of repair explained by the age of the machine.

A high R-squared suggests that the age of the machine is a good predictor of the monthly cost of repair.

d. Conduct a test to determine whether the age of a machine and its monthly cost of repair are linearly related:

To test the linearity of the relationship between the age of the machine and its monthly cost of repair, you can use the coefficient of correlation (r) derived from the regression analysis. The value of r will range from -1 to 1, with 1 indicating a perfect positive linear relationship.

You can conduct a hypothesis test to determine if the correlation coefficient is statistically significant. This can be done using a t-test, where the null hypothesis is that there is no linear relationship between the two variables. If the p-value is less than the significance level (e.g., alpha = 0.05), we can reject the null hypothesis and conclude that there is a significant linear relationship between the age of the machine and monthly repair cost.

e. Determine if the fit of the simple linear model is good enough to predict the monthly repair cost of a welding machine that is 120 months old:

To determine if the simple linear model is good enough to predict the monthly repair cost of a welding machine that is 120 months old, you can use the regression equation obtained in step 'a'.

Plug in the value of 120 for the age of the machine in the equation. The predicted monthly repair cost will be the estimated value based on the regression analysis.

To calculate the prediction interval, you need to consider the uncertainty in the estimate. A 95% prediction interval takes into account the variability of the data and gives you a range of values within which you can expect the actual monthly repair cost for a machine of particular age.

The prediction interval can also be calculated using statistical software or Excel. It is wider than a simple confidence interval because it includes both the uncertainty in the estimate of the regression line (confidence interval) and the random variability of the data.

If the prediction interval is narrow, it suggests that the model is good enough to make predictions. However, if the prediction interval is wide, it indicates that there is a lot of variability in the data, and the model may not be accurate enough for reliable predictions.