What type of model would you use if you wanted to find the relationship between a set of variables 1 point?

The linear regression model is:

\(\text{price}=\beta_0+\beta_1\text{age}+\epsilon\)

To test whether age is a statistically significant negative linear predictor of price, we can set up the following hypotheses:.

\(H_0\colon \beta_1=0\)

\(H_a\colon \beta_1< 0\)

We need to verify that our assumptions are satisfied. Let's do this in Minitab. Remember, we have to run the linear regression analysis to check the assumptions.

Assumption 1: Linearity

The scatterplot below shows that the relationship between age and price scores is linear. There appears to be a strong negative linear relationship and no obvious outliers.

Assumption 2: Independence of errors

There does not appear to be a relationship between the residuals and the fitted values. Thus, this assumption seems valid.

Assumption 3: Normality of errors

On the normal probability plot we are looking to see if our observations follow the given line. This graph does not indicate that there is a violation of the assumption that the errors are normal. If a probability plot is not an option we can refer back to one of our first lessons on graphing quantitative data and use a histogram or boxplot to examine if the residuals appear to follow a bell shape.

Assumption 4: Equal Variances

Again we will use the plot of residuals versus fits. Now we are checking that the variance of the residuals is consistent across all fitted values. This assumption seems valid.

Model Summary

Coefficients

Regression Equation

price = 7850 - 485.0 age

From the output above we can see that the p-value of the coefficient of age is 0.000 which is less than 0.001. The Minitab output is for a two-tailed test and we are dealing with a left-tailed test. Therefore, the p-value for the left-tailed test is less than \(\frac{0.001}{2}\) or less than 0.0005.

We can thus conclude that age (in years) is a statistically significant negative linear predictor of price for any reasonable \(\alpha\) value.

\(\beta_0\) is the y-intercept, which means it is the value of price when age is equal to 0. It is possible for a vehicle to have number of years equal to 0. Therefore, it does have an interpretable meaning. We should use caution if we use this model to predict the price of a car with age equal to 0 because it is outside the range of values used to estimate the model.

The 95% confidence interval for the population slope is:

\(\hat{\beta}_1\pm t_{\alpha/2}\text{SE}(\hat{\beta}_1)\)

Using the output, \(\hat{\beta}_1=-485\) and the \(\text{SE}(\hat{\beta}_1)=43.9\). We need to have \(t_{\alpha/2}\) with \(n-2\) degrees of freedom. In this case, there are 18 observations so the degrees of freedom are \(18-2=16\). Using software, we find \(t_{\alpha/2}=2.12\).

The 95% confidence interval is:

\(-485\pm 2.12(43.9)\)

\((-578.068, -391.932)\)

We are 95% confident that the population slope for the regression model is between -578.068 and -391.932. In other words, we are 95% confident that, for every one year increase in age, the price of a vehicle will decrease between 391.932 and 578.068 dollars.

We can use the regression equation with \(\text{age}=7\):

\(price=7850-485(7)=4455\)

We can expect the price to be $4455.

The residual standard error is estimated by s, which is calculated as:

\(s=\sqrt{\text{MSE}}=\sqrt{253156}=503.146\)

Note! The MSE is found in the ANOVA table that is part of the regression output in Minitab.

It is also shown as \(s\) under the model summary in the output.

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