Ch 12.5 Prediction

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Ch 12.5 Prediction

Criteria for using the line of best fit to predict y:

1) Scatte plot indicates a linear pattern with no other non-linear patterns or outliers.

2) (x, y) are matched-pair and linearly correlated. Scatter plot does not show non-linear patterns.

3) x is within the prediction domain for intrapolation. The range of x values in the sample is the appropriate domain.

4) For each fixed value of x, the corresponding values of y have a normal distribution. (loose requirement)

Find best prediction for y.

Step1: Find the Linear regression equation, p-value and r and the scatter plot.

Enter matched pair data to statdisk columns. Use Analysis/Correlation and Regression/ Enter significance, select data columns.

Output: r, critical r, p-value, b0 and b1, r² and scatter plot. \( \hat{y} = b_0 + b_1 x \) is the regression line.
Step 2: check scatter plot linear pattern and inspect if there is any non-linear pattern or outliers.

Step 3: Determine if (x, y) are linear related.

If p-value ≤ α , reject H0, conclude x, y are linear correlated

If p-value > α, conclude x, y are not linear correlated. OR

If r outside the range of -critical r and +critical r, conclude x, y are linear correlated.

Step 4: Find the best predicted y.

If x,y are linear correlated, use the linear regression equation to find the best predicted y, .

\( \hat{y} = b_0 + b_1 x \)

If x, y are not linear correlated, use \( \bar{y} \) (mean of y) as best predicted y.

To find \( \bar{y} \), use Statdisk/ Explore Data/ to find mean of y.

How good is the prediction

The correlation of determination, r² descibes how good the linear regression is in predicating the variation of y. The higher the correlation of determination, the better is the prediction.

Ex 1:
Given the following matched pair data:

matched pair data

Use the information to find the best predicted value of sleep time if the screen time is 3 hours. Use α=0.05

1) Find regression line equation, p-value,r, scatter plot.

Enter data to 2 columns of statdisk. Use Analysis/Correlation and Regression/ Enter significance, select data columns.

Output: r = -0.579, critical r = ±0.878, p-value=0.3061, b0= 9.774, b1=-1.099, scatter plot in other tab.

Linear regression equation is \( \hat{y} = 9.774 + 1.099 x \)

2) Check Scatter plot

scatter plot There is no systematic non-linear pattern. There seems to be a negative weak correlation.

3) Determine if x, y are linear correlated.

since r = -0.579 is between -0.878 and +0.878, conclude no linear correlation.

OR : since p-value (0.306) > 0.05, conclude no linear correlation.

4) Since x, y are not linearly correlated, the best predicted value is mean of y. \( \bar{y} \)

Statdisk/Data/Explore data/select sleep time column, mean = 7.4. So best predicted y when x = 3 hour is 7.4 hours

Ex 3. Given matched pair data for 7 students’ study hour and final exam scores.

matched pair data

Use α= 0.05 to predict a student’s final score based on study hour of 6 hours.

Step 1) Find linear regression equation, p-value ,r and scatter plot.

Enter study hour and final scores to statdisk.

Analysis/Correlation and Regression/enter significance = 0.05, select data columns.

Output: r = 0.789, critical r = ±0.754, p-value = 0.0351, b0 =64.017, b1 = 3.217.

So \( \hat{y} = 64.017 + 3.217 x \) is the linear regression line.

Step 2) Check scatter plot:

Check scatter plot tab.

There is no non-linear patterns or outliers. The correlation is not very strong.

Step 3) Determine if x and y are linearly correlated.

Since p-value < 0.05 reject H0, conclude x and y are linearly correlated. OR r = 0.789 is outside the range of -0.754 and +0.754, so there is linear correlation.

4) Since x, y are linearly correlated, use linear regression line to find best prediction of score when x = 6 hours.

\( \hat{y} = 64.017 + 3.217 * 6 \). Best predicted scores = 83.3

b) Can the line of best fit equation be used to find predicted scores when study hour is 0. Explain.

Since 0 is not within the prediction domain, the regression line should not be used for prediction.

Ex4:

Given x, y are matched pair data with no non-linear pattern in scatter plot with =3.3 and

The line of best fit is \( \hat{y} = 2.1 + 0.32 x \) .

Correlation r = 0.82, and critical r= 0.754, find the best predicted y when x is 2.5 at α= 0.05.

Since r = 0.82 > 0.754 so x, y are linear correlated.

The best predicted y is from the linear regression line = 2.1+ 0.32(2.5) =2.9.

Ex5:

Given x, y are matched pair data with no non-linear pattern in scatter plot with \( \bar{x}=6.5 \) and \( \bar{y} = 1.9 \).

The line of best fit is \( \hat{y} = 2.1 + 0.32 x \).

Given correlation p-value = 0.11, find the best predicted y when x is 5 at α= 0.05.

Since p-value 0.11 > 0.05 so x, y are not linear correlated.

The best predicted y is the mean of y = 1.9 instead of using the linear regression line.