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13.40 An experiment was carried out by geologists to see how the time necessary to drill a distance of 5 feet in rock \((y\), in minutes) depended on the depth at which the drilling began \((x\), in feet, between 0 and 400 ). We show part of the Minitab output obtained from fitting the simple linear regression model ("Mining Information. The regression equation is Time \(=4.79+0.0144\) depth \(\begin{array}{lrrrr}\text { Predictor } & \text { Coef } & \text { Stdev } & \text { t-ratio } & p \\ \text { Constant } & 4.7896 & 0.6663 & 7.19 & 0.000 \\\ \text { depth } & 0.014388 & 0.002847 & 5.05 & 0.000 \\ s=1.432 & \text { R-sq }=63.0 \% & R-s q(a d j)=60.5 \% & \end{array}\) Analysis of Variance \(\begin{array}{lrrrrr}\text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & p \\ \text { Regression } & 1 & 52.378 & 52.378 & 25.54 & 0.000 \\ \text { Error } & 15 & 30.768 & 2.051 & & \\ \text { Total } & 16 & 83.146 & & & \end{array}\) a. What proportion of observed variation in time can be explained by the simple linear regression model? b. Does the simple linear regression model appear to be useful? c. Minitab reported that \(s_{a+b(200)}=.347 .\) Calculate a \(95 \%\) confidence interval for the mean time when depth \(=200\) feet. d. A single observation on time is to be made when drilling starts at a depth of 200 feet. Use a \(95 \%\) prediction interval to predict the resulting value of time.

Step by step solution

01

Calculate Proportion of Observed Variation

The proportion of observed variation can be calculated by squaring the correlation coefficient \(R\), also known as \(R^2\) or the coefficient of determination. In this case, this value is already given as 63%. That means that 63% of the time change is explained by the depth of the drilling.

02

Determine Usefulness of the Regression Model

The usefulness of a regression model is generally estimated by looking at the P-value and the R-squared value. In this case, both p-values are given as 0.000, being less than 0.05, hence, the depth of the drilling is a significant predictor of the time needed. Additionally, \(R^2\) which represents the proportion of the variance for a dependent variable that's explained by an independent variable is 63%, which indicates a good level of explanation and thus, the regression model appears to be useful.

03

Calculate the 95% Confidence Interval

To calculate the confidence interval, we first need to plug the depth value (200) into the regression equation which is Time = 4.79 + 0.0144 x Depth. Then, we need to multiply the standard deviation \(s_{a+b(200)}=0.347\) by the value of the t statistic for the desired level of confidence and degrees of freedom. After that, we subtract and add this value to the mean calculated by the regression equation. This will yield the lower and upper limit of the confidence interval.

04

Calculate the 95% Prediction Interval

The prediction interval for a single observation is obtained in a similar way as the confidence interval, but with an extra term that adjusts for the variability of individual data points around regression line. We perform a similar process of plugging in the depth value and calculating the mean as above, but we multiply the \(s_{a+b(200)}=0.347\) by a slightly larger t value (for the same level of confidence and degrees of freedom) to get the added prediction interval variability. We then subtract and add this value to the mean as we did above to get the 95% prediction interval.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coefficient of Determination

The Coefficient of Determination is symbolized as \( R^2 \) and provides a measure of how well the regression line fits the data. It represents the proportion of variation in the dependent variable that can be explained by the independent variable. In the given experiment, \( R^2 \) is 63%. This means that 63% of the variation in drilling time can be attributed to differences in starting depth.

This metric helps assess the efficiency of models in making predictions. A higher \( R^2 \) indicates a better fit, implying that our model is performing well in explaining the data. However, it's important to note that a high \( R^2 \) alone does not imply that the model is perfect or flawless. The context of the problem and other statistical indicators (such as residual plots and p-values) should also be considered when assessing the goodness of fit.

P-value in Regression

When conducting regression analysis, P-values help us determine the significance of our predictors. A P-value indicates the probability that the results observed were actually due to chance. In general, a P-value less than 0.05 is considered statistically significant. In this exercise, the P-value for both the intercept and the slope (depth) is given as 0.000.

This means that, for the purposes of this experiment, the chances of the observed correlation between depth and time being due to randomness are practically zero. Thus, we can conclude that depth is a significant factor in determining drilling time.

By using P-values, researchers can confidently assess which variables they should pay attention to when building their regression models. It reinforces the idea that our model is statistically sound and helps to support the usefulness of the regression analysis.

Confidence Interval

Confidence Intervals give us a range within which we expect the true mean value of the dependent variable to lie, given a particular value of the independent variable. For a depth of 200 feet in this exercise, we calculate a 95% confidence interval for the mean drilling time.

To do this, we first substitute the depth value into our regression equation: \( ext{Time} = 4.79 + 0.0144 imes 200 \). Then, the standard error \( s_{a+b(200)}=0.347 \) is used alongside the appropriate t-distribution value to calculate the interval. The outcome encompasses our predicted mean with 95% confidence.

Such intervals are widely used in statistical studies because they provide valuable information about the reliability and stability of the predictions. However, users should remember that these intervals do not cover individual observations but rather the average outcome if the experiment were repeated numerous times.

Prediction Interval

Prediction Intervals provide an estimated range for individual future observations based on our model. Unlike confidence intervals, they account for the variability inherent in individual measurements. This is crucial when you're trying to predict a single observation.

For a depth starting at 200 feet, a 95% prediction interval would be calculated similarly to the confidence interval but includes additional variance to cater for individual prediction error. Hence, the interval is wider to accommodate this additional uncertainty.

To obtain this interval, you need to calculate the mean time like in the confidence interval, but multiply \( s_{a+b(200)}=0.347 \) by a slightly larger t-value to achieve the broader range.

Prediction intervals are particularly useful in practice when we want to make forecasts or predictions about specific instances. They provide a realistic assessment by considering both the model's error and the natural variability of the data, thus enabling more informed decision-making.