1. Explain the regression model and the regression model assumptions.
2. Perform the hypothesis test for the slope of the population regression equation.
3. Construct confidence intervals for the slope .
4. Use confidence intervals to perform the hypothesis test for the slope .

1. Construct and interpret a scatterplot of the response variable versus the predictor variable .
2. Calculate and interpret the correlation coefficient .
3. Compute the regression equation . Interpret the meaning of the intercept and the slope of the regression equation.
4. Predict the number of text messages sent by a 20-year-old person, and calculate the prediction error (residual).

1. The number of messages depends on age, and not vice versa, so the predictor variable is age and the response variable is messages. Also, note that in (d) we are trying to predict the number of text messages, which tells us that messages is the response variable because we never try to predict the known value of . The TI-83/84 scatterplot is shown in Figure 1. As age increases, the number of messages tends to decrease.

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   Figure 13.1: FIGURE 1 TI-83/84 scatterplot of messages versus age.
2. Figure 2 shows the correlation coefficient , calculated by the TI-83/84. Age and messages are negatively correlated. An increase in age is associated with a decrease in the number of messages.

3. Figure 2 shows that and , and thus the regression equation is

   We can interpret and as follows:

   • The intercept is the estimated number of text messages sent by someone aged , which does not make sense because this value lies far below the minimum value of and therefore represents extreme extrapolation.
   • The slope means there is an estimated decrease of 1.5 in the number of text messages for each additional year of age.
   

   Figure 13.2: FIGURE 2 TI-83/84 correlation and regression results.

4. For a 20-year-old person, the estimated number of daily text messages is

   The actual number of text messages sent by our 20-year-old in Table 1 is . Our prediction from (c) is . Thus, our prediction error (or residual) is: . Our 20-year-old sent slightly fewer text messages than expected.

YOUR TURN #1

1. Construct and interpret a scatterplot of the response variable versus the predictor variable .
2. Calculate and interpret the correlation coefficient .
3. Compute the regression equation . Interpret the meaning of the intercept and the slope of the regression equation.
4. Predict the score for a 22-year old person, and calculate the prediction error (residual).

Note that there is no “hat” on the in the population regression equation because the equation represents a model of the relationship between the actual values of and , not an estimate of .

1. Zero-mean assumption. The error term is a random variable, with a mean of 0. That is, the expected value of the random variable is .
2. Constant variance assumption. The variance of , which is denoted as , is the same regardless of the value of .
3. Independence assumption. The values of are independent of each other.
4. Normality assumption. The error term is a normal random variable.

Developing Your Statistical Sense

1. Calculate the residuals .
2. Verify the regression assumptions.

1. Table 2 contains the and data from Table 1, the fitted (predicted) values , and the residuals .
   Table 13.3: Table 2 Calculating the residuals

   		Fitted (predicted) values	Residuals
   18	35	33.6	1.4
   20	29	30.6	−1.6
   22	27	27.6	−0.6
   24	28	24.6	3.4
   26	19	21.6	−2.6
   28	16	18.6	−2.6
   30	19	15.6	3.4
   32	12	12.6	−0.6
   34	8	9.6	−1.6
   36	8	6.6	1.4

2. The scatterplot in Figure 5 of the residuals versus fitted values shows no strong evidence of the unhealthy patterns shown in Figure 4. Thus, the independence assumption, the constant variance assumption, and the zero-mean assumption are verified. Also, the normal probability plot of the residuals in Figure 6 indicates no evidence of departures from normality in the residuals. Therefore, we conclude that the regression assumptions are verified.
   

   Figure 13.5: FIGURE 5 Scatterplot of residuals versus fitted values.
   

   Figure 13.6: FIGURE 6 Normal probability plot of the residuals.

YOUR TURN #2

1. Calculate the residuals . for the regression of score on age.
2. Verify the regression assumptions.

Here, refers to the standard error of the estimate, not the sample standard deviation.

1. Find , the slope of the regression line.
2. Calculate , the standard error of the estimate.
3. Compute , the numerator of the sample variance of the data.

1. From Example 1, the slope of the regression line is .

2. Recall from Section 4.3 (page 228) that

   is the standard error of the estimate. Squaring each residual from Table 2 gives us the squared residuals in Table 3, and the sum of squared residuals, or sum of squares error, equal to

   Then the standard error of the estimate is

   Table 13.4: Table 3 Calculating SSE

   Residuals	Squared residuals
   1.4	1.96
   −1.6	2.56
   −0.6	0.36
   3.4	11.56
   −2.6	6.76
   −2.6	6.76
   3.4	11.56
   −0.6	0.36
   −1.6	2.56
   1.4	1.96

3. To compute , we note from page 110 in Chapter 3 that the sample variance of is

   Multiplying each side of the equation by , we obtain an equation for the quantity :

   The TI-83/84 output from Figure 7 shows that , and, because ,

   Therefore,

   

   Figure 13.7: FIGURE 7 Summary statistics for the (age) data.

• Step 1 State the hypotheses.
  • No linear relationship exists between age and text messages.
  • A linear relationship exists between age and text messages.

• Step 2 Find the critical value and the rejection rule. To find , use the distribution table (Table D in the Appendix) for a two-tailed test and degrees of freedom . The rejection rule for this two-tailed test is

  Here, , so . For level of significance , the table gives us . We will reject if .

• Step 3 Calculate . From Example 3, we have

• Step 4 State the conclusion and the interpretation. Because , we reject . There is evidence, at level of significance , that and that a linear relationship exists between age and text messages.

• Step 1 State the hypotheses and the rejection rule.

  • No linear relationship exists between time and score.
  • A linear relationship exists between time and score.

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  Figure 13.8: FIGURE 8 Residuals versus fitted values plot.
  

  Figure 13.9: FIGURE 9 Normal probability plot of the residuals.

  The rejection rule is: reject if the .

• Step 2 Calculate .

  From page 226 in Section 4.3, we have . From Example 13 in Chapter 4 on page 228, we have

  From the TI-83/84 summary statistics, we have the standard deviation of the (time) data to be . Thus, using the relationship we learned in Example 3:

  Therefore,

  

  TI-83/84 summary statistics for (time) data.

• Step 3 Find the -value. For instructions, see the Step-by-Step Technology Guide on page 730. The regression results (including the -value) for the TI-83/84, Excel, Minitab, and CrunchIt! are shown in Figures 10, 11, 12, and 13. (Differing results are due to rounding.)

  

  Figure 13.10: FIGURE 10 TI-83/84 regression results.

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  Figure 13.11: FIGURE 11 Excel regression result.
  

  Figure 13.12: FIGURE 12 Minitab regression results.
  

  Figure 13.13: FIGURE 13 Crunchit! regression results.

• Step 4 The -value of about , so we reject . Evidence exists, at level of significance , for a linear relationship between time and score.

YOUR TURN #3

• ,
• , and
• .

• The margin of error means that, when we repeatedly take samples from this population, most of the time the sample estimate will be within of the unknown value of the slope of the population regression line.
• We are 95% confident that the interval captures the slope of the population regression line.
• Because is the increase in memory-test score per added minute of memorization, we are 95% confident that, for each additional minute of memorization, the increase in memory-test score will lie between 1.6157 and 2.3843 points.

• If a confidence interval for does not contain zero, then we would reject for level of significance , and conclude that a linear relationship exists between and .
• If a confidence interval for does contain zero, then we would not reject for level of significance .

1. Construct and interpret a 99% confidence interval for the slope for the text messaging data in Table 1.
2. Use the confidence interval in (a) to test whether a linear relationship exists between age and text messages, using level of significance .

1. The regression assumptions were verified in Example 2. Also,
   • In Example 1, we found .
   • In Example 3, we calculated , and .

   From the table, we find that, for 99% confidence, for degrees of freedom is . So, our margin of error si

   The 99% confidence interval for is then given by

   We are 99% confident that the interval (−1.9448, −1.0552) captures the slope of the population regression line. That is, we are 99% confident that, for each additional year of age, the decrease in the number of text messages lies between 1.9448 and 1.0552.

2. The hypotheses are

   No linear relationship exists between age and text messages.

   A linear relationship exists between age and text messages.

Figure 13.14: FIGURE 14 Scatterplot of Scrabble frequency versus English relative frequency of letters, with regression line.

Figure 13.15: FIGURE 15 Minitab regression output.

• No linear relationship exists between Scrabble frequency and English relative frequency.
• A linear relationship exists between Scrabble frequency and English relative frequency.

Figure 13.16: FIGURE 16 Scatterplot of Scrabble frequency versus English relative frequency. Linear regression would not be appropriate.

• What is a “good” Scrabble tile to pick up? In the best case, it would be a letter with high English frequency worth lots of Scrabble game points. Unfortunately, the two do not go together. The high-frequency letters such as E and T have low point values, and the high point-value letters such as Q and Z have low frequencies. But we can still make comparisons.
• Which would you rather pick up, a D or a G? It would seem that D would be preferable because it has the same point value as G with much higher English frequency.
• Which do you prefer between J and X? They are worth the same points, but X has a higher frequency in English, so it is easier to make words with it in the game. The letter H seems to have a good combination of high points and moderate frequency.