1 Parameters, Statistics, and Statistical Inference
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Chapter 14
The calculation of the estimate of standard deviation is based on the sum of
the squared residuals for the sample. This quantity is called the sum of squared
errors and is denoted by SSE. Synonyms for “sum of squared errors” are residual sum of squares or sum of squared residuals. To ﬁnd the SSE, residuals are
calculated for all observations, then the residuals are squared and summed.
The standard deviation from the regression line is
sϭ
B
Sum of squared residuals
SSE
ϭ
nϪ2
Bn Ϫ 2
and this sample statistic estimates the population standard deviation s.
Estimating the Standard Deviation for a Simple Regression Model
formula
The formula for estimating the standard deviation for a simple regression
model is
SSE ϭ a 1 yi Ϫ yˆi 2 2 ϭ a e 2i
g 1yi Ϫ yˆi 2 2
SSE
sϭ
ϭ
Bn Ϫ 2 B n Ϫ 2
The statistic s is an estimate of the population standard deviation s.
Technical Note: Notice the difference between the estimate of s in the regression situation compared to what it would be if we simply had a random sample
of the yi’s without information about the xi’s:
Sample of y’s only:
sϭ
g 1 yi Ϫ y2 2
B
nϪ1
Sample of 1x, y2 pairs, linear regression:
sϭ
g 1 yi Ϫ yˆi 2 2
B
nϪ2
Remember that in the regression context, s is the standard deviation of the y
values at each x, not the standard deviation of the whole population of y values.
Example 14.3
Relationship Between Height and Weight for College Men Figure 14.4
displays regression results from the Minitab program and a scatterplot for
the relationship between y ϭ weight (pounds) and x ϭ height (inches) in a
The regression equation is
Weight = –318 + 7.00 Height
Predictor
Constant
Height
Coef
–317.9
6.996
S = 24.00
R-Sq = 32.3%
240
SE Coef
110.9
1.581
T
–2.87
4.42
R-Sq(adj) = 30.7%
P
0.007
0.000
220
Weight (lb)
606
200
180
160
140
66
67 68
69
70 71 72 73 74 75
Height (in.)
Figure 14.4 ❚ The relationship between weight and height for n ؍43 college men
Inference About Simple Regression
Watch a video example at http://
1pass.thomson.com or on your CD.
in summary
607
sample of n ϭ 43 men in a statistics class. The regression line for the sample is
yˆ ϭ Ϫ318 ϩ 7x, and this line is drawn onto the plot. We see from the plot that
there is considerable variation from the line at any given height. The standard
deviation, shown in the last row of the computer output to the left of the plot, is
“S ϭ 24.00.” This value roughly measures, for any given height, the general size
of the deviations of individual weights from the mean weight for the height.
The standard deviation from the regression line can be interpreted in conjunction with the Empirical Rule for bell-shaped data stated in Section 2.7. Recall, for instance, that about 95% of individuals will fall within 2 standard deviations of the mean. As an example, consider men who are 72 inches tall. For
men with this height, the estimated average weight determined from the regression equation is Ϫ318 ϩ 7.00(72) ϭ 186 pounds. The estimated standard deviation from the regression line is s ϭ 24 pounds, so we can estimate that about 95%
of men 72 inches tall have weights within 2 ϫ 24 ϭ 48 pounds of 186 pounds,
which is 186 Ϯ 48, or 138 to 234 pounds. Think about whether this makes sense
for all the men you know who are 72 inches (6 feet) tall. ■
Interpreting the Standard Deviation for Regression
The standard deviation for regression estimates the standard deviation of the
differences between values of y and the regression equation that relates the
mean value of y to x. In other words, it measures the general size of the differences between actual and predicted values of y.
t h o u g h t q u e s t i o n 1 4 . 1 Regression equations can be used to predict the value of a response
variable for an individual. What is the connection between the accuracy of predictions based on a particular regression line and the value of the standard deviation
from the line? If you were deciding between two different regression models for predicting the same response variable, how would your decision be affected by the relative values of the standard deviations for the two models?*
The Proportion of Variation Explained by x
In Chapter 5, we learned that the squared correlation r 2 is a useful statistic. It
is used to measure how well the explanatory variable explains the variation in
the response variable. This statistic is also denoted as R 2 (rather than r 2), and
the value is commonly expressed as a percentage. Researchers typically use the
phrase “proportion of variation explained by x” in conjunction with the value
of r 2. For example, if r 2 ϭ .60 (or 60%), the researcher may write that the explanatory variable explains 60% of the variation in the response variable.
*H I N T :
Read the ﬁrst paragraph of this section (p. 605).
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Chapter 14
The formula for r 2 presented in Chapter 5 was
r2 ϭ
SSTO Ϫ SSE
SSTO
The quantity SSTO is the sum of squared differences between observed y values
and the sample mean y . It measures the size of the deviations of the y values from the overall mean of y, whereas SSE measures the deviations of the
y values from the predicted values of y.
Example 14.4
R 2 for Heights and Weights of College Men In Figure 14.4 for Example 14.3
(p. 606), we can ﬁnd the information “R-sq ϭ 32.3%” for the relationship between weight and height. A researcher might write “the variable height explains
32.3% of the variation in the weights of college men.” This isn’t a particularly
impressive statistic. As we noted before, there is substantial deviation of individual weights from the regression line, so a prediction of a college man’s weight
based on height may not be particularly accurate. ■
t h o u g h t q u e s t i o n 1 4 . 2 Look at the formula for SSE, and explain in words under what condition SSE ϭ 0. Now explain what happens to r 2 when SSE ϭ 0, and explain whether
that makes sense according to the deﬁnition of r 2 as “proportion of variation in y explained by x.” *
Example 14.5
Driver Age and Highway Sign-Reading Distance In Example 5.2 (p. 153),
we examined data for the relationship between y ϭ maximum distance (feet)
at which a driver can read a highway sign and x ϭ the age of the driver. There
were n ϭ 30 observations in the dataset. Figure 14.5 displays Minitab regression
Watch a video example at http://
1pass.thomson.com or on your CD.
For software help, download your
Minitab, Excel, TI-83, SPSS, R, and
JMP manuals from http://1pass
.thomson.com, or ﬁnd them on
your CD.
The regression equation is
Distance = 577 – 3.01 Age
Predictor
Constant
Age
S = 49.76
Coef
576.68
–3.0068
SE Coef
23.47
0.4243
R-Sq = 64.2%
T
24.57
–7.09
P
0.000
0.000
R-Sq (adj) = 62.9%
Analysis of Variance
Source
Regression
Residual Error
Total
DF
1
28
29
SS
124333
69334
193667
Unusual Observations
Obs
Age
Distance
27
75.0
460.00
MS
124333
2476
Fit
351.17
SE Fit
13.65
F
50.21
P
0.000
Residual
108.83
St Resid
2.27R
R denotes an observation with a large standardized residual
Figure 14.5 ❚ Minitab output: Sign-reading distance and driver age
*H I N T :
Remember that SSE stands for “sum of squared errors.” The formula for r 2 is given just before Example 14.4.
Inference About Simple Regression
609
output for these data. The equation describing the linear relationship in the
sample is
Average distance ϭ 577 Ϫ 3.01 ϫ Age
From the output, we learn that the standard deviation from the regression line
is s ϭ 49.76 and R-sq ϭ 64.2%. Roughly, the average deviation from the regression line is about 50 feet, and the proportion of variation in sign-reading distances explained by age is .642 or 64.2%.
The analysis of variance table provides the pieces needed to compute r 2
and s:
SSE ϭ 69,334
sϭ
69,334
SSE
ϭ
ϭ 49.76
B n Ϫ 2 B 28
SSTO ϭ 193,667
SSTO Ϫ SSE ϭ 193,667 Ϫ 69,334 ϭ 124,333
14.2 Exercises are on page 626.
r2 ϭ
124,333
SSTO Ϫ SSE
ϭ
ϭ .642, or 64.2% ■
SSTO
193,667
14.3 Inference About the Slope
of a Linear Regression
In this section, we will learn how to carry out a hypotheses test to determine
whether we can infer that two variables are linearly related in the larger population represented by a sample. We will also learn how to use sample regression
results to calculate a conﬁdence interval estimate of a population slope.
Hypothesis Test for a Population Slope
The statistical signiﬁcance of a linear relationship can be evaluated by testing whether or not the population slope is 0. If the slope is 0 in a simple linear regression model, the two variables are not related because changes in the
x variable will not lead to changes in the y variable. The usual null hypothesis
and alternative hypothesis about b1, the slope of the population regression line
E(Y ) ϭ b0 ϩ b1x, are
H0: b1 ϭ 0
Ha: b1 0
(the population slope is 0, so y and x are not linearly related )
(the population slope is not 0, so y and x are linearly related )
The alternative hypothesis may be one-sided or two-sided, although most statistical software uses the two-sided alternative.
The test statistic used to do the hypothesis test is a t-statistic with the same
general format that we used in Chapters 12 and 13. That format, and its application to this situation, is
tϭ
Sample statistic Ϫ Null value
b1 Ϫ 0
ϭ
Standard error
s.e.1b1 2
This is a standardized statistic for the difference between the sample slope and
0, the null value. Notice that a large value of the sample slope (either positive or
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Chapter 14
negative) relative to its standard error will give a large value of t. If the mathematical assumptions about the population model described in Section 14.1 are
correct, the statistic has a t-distribution with n Ϫ 2 degrees of freedom. The
p-value for the test is determined using that distribution.
It is important to be sure that the necessary conditions are met when using
any statistical inference procedure. The necessary conditions for using this test,
and how to check them, will be discussed in Section 14.5.
“By hand” calculations of the sample slope and its standard error are cumbersome. Fortunately, the regression analysis of most statistical software includes a t-statistic and a p-value for this signiﬁcance test.
formula
Formula for the Sample Slope and Its Standard Error
In case you ever need to compute the values by hand, here are the formulas for
the sample slope and its standard error:
b1 ϭ r
sy
sx
s.e.1b1 2 ϭ
s
2g 1x1 Ϫ x2
2
where s ϭ
SSE
Bn Ϫ 2
In the formula for the sample slope, sx and sy are the sample standard deviations
of the x and y values, respectively, and r is the correlation between x and y.
Example 14.6
For software help, download your
Minitab, Excel, TI-83, SPSS, R, and
JMP manuals from http://1pass
.thomson.com, or ﬁnd them on
your CD.
Hypothesis Test for Driver Age and Sign-Reading Distance Figure 14.5
(p. 608) for Example 14.5 presents Minitab output for the regression of signreading distance ( y) and driver age. The part of the output that is used to test the
statistical signiﬁcance of the observed relationship is shown in bold. This line of
output gives values for the sample slope, the standard error of the sample slope,
the t-statistic, and the p-value for the test of
H0: b1 ϭ 0
Ha: b1 0
(the population slope is 0, so y and x are not linearly related )
(the population slope is not 0, so y and x are linearly related )
The test statistic is
tϭ
Sample statistic Ϫ Null value
b1 Ϫ 0
Ϫ3.0068 Ϫ 0
ϭ
ϭ
ϭ Ϫ7.09
Standard error
s.e.1b1 2
0.4243
The p-value (underlined in the output) is given to three decimal places as .000.
This means that the probability is virtually 0 that the sample slope could be as
far from 0 or farther than it is if the population slope really is 0. Because the pvalue is so small, we can reject the null hypothesis and infer that the linear relationship observed between the two variables in the sample represents a real relationship in the population. ■
technical note
Most statistical software reports a p-value for a two-sided alternative hypothesis when doing a test for whether the slope in the population is 0. It
may sometimes make sense to use a one-sided alternative hypothesis instead. In that case, the p-value for the one-sided alternative is (reported p>2)
if the sign of b1 is consistent with Ha, but is 1 Ϫ (reported p>2) if it is not.
Inference About Simple Regression
611
Conﬁdence Interval for the Population Slope
The signiﬁcance test of whether or not the population slope is 0 tells us only
whether we can declare the relationship to be statistically signiﬁcant. If we decide that the true slope is not 0, we might ask, “What is the value of the slope?”
We can answer this question with a conﬁdence interval for b1, the population
slope.
The format for this conﬁdence interval is the same as the general format
used in Chapters 10 and 11, which is
Sample statistic Ϯ Multiplier ϫ Standard error
The sample statistic is b1, the slope of the least-squares regression line for the
sample. As has been shown already, the standard error formula is complicated,
and we will usually rely on statistical software to determine this value. The
“multiplier” will be labeled t * and is determined by using a t-distribution with
df ϭ n Ϫ 2. Table A.2 can be used to ﬁnd the multiplier for the desired conﬁdence level.
formula
Formula for Conﬁdence Inter val for B 1 , the Population Slope
A conﬁdence interval for b1 is
b1 Ϯ t * s.e.(b1)
The multiplier t * is found by using a t-distribution with n Ϫ 2 degrees of freedom and is such that the probability between Ϫt * and ϩt * equals the conﬁdence level for the interval.
Example 14.7
95% Conﬁdence Interval for Slope Between Age and Sign-Reading Distance In Figure 14.5 (p. 608), we see that the sample slope is b1 ϭ Ϫ3.01 and
s.e.(b1) ϭ 0.4243. There are n ϭ 30 observations, so df ϭ 28 for ﬁnding t *. For a
95% conﬁdence level, t * ϭ 2.05 (see Table A.2). The 95% conﬁdence interval for
the population slope is
Ϫ3.01 Ϯ 2.05 ϫ 0.4243
Ϫ3.01 Ϯ 0.87
Ϫ3.88 to Ϫ2.14
With 95% conﬁdence, we can estimate that in the population of drivers represented by this sample, the mean sign-reading distance decreases somewhere
between 2.14 and 3.88 feet for each one-year increase in age. ■
t h o u g h t q u e s t i o n 1 4 . 3 In previous chapters, we learned that a conﬁdence interval can be
used to determine whether a hypothesized value for a parameter can be rejected.
How would you use a conﬁdence interval for the population slope to determine
whether there is a statistically signiﬁcant relationship between x and y? For example,
why is the interval that we just computed for the sign-reading example evidence that
sign-reading distance and age are related?*
*H I N T :
What is the null value for the slope? Section 13.5 discusses the connection between
conﬁdence intervals and signiﬁcance tests.
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Chapter 14
SSPS t i p
Calculating a 95% Conﬁdence Inter val for the Slope
●
Use AnalyzebRegressionbLinear Regression. Specify the y variable in
the Dependent box and specify the x variable in the Independent(s) box.
●
Click Statistics and then select “Conﬁdence intervals” under “Regression
Coefﬁcients.”
Testing Hypotheses About
the Correlation Coefﬁcient
In Chapter 5, we learned that the correlation coefﬁcient is 0 when the regression
line is horizontal. In other words, if the slope of the regression line is 0, the correlation is 0. This means that the results of a hypothesis test for the population
slope can also be interpreted as applying to equivalent hypotheses about the
correlation between x and y in the population.
We use different notation to distinguish between a correlation computed
for a sample and a correlation within a population. It is commonplace to use
the Greek letter r (pronounced “rho”) to represent the correlation between two
variables within a population. Using this notation, null and alternative hypotheses of interest are as follows:
H0 : r ϭ 0
Ha: r 0
(x and y are not correlated)
(x and y are correlated)
The results of the hypothesis test described before for the population slope b1
can be used for these hypotheses as well. If we reject H0: b1 ϭ 0, we also reject
H0: r ϭ 0. If we decide in favor of Ha: b1 0, we also decide in favor of Ha: r 0.
Many statistical software programs, including Minitab, will give a p-value
for testing whether the population correlation is 0 or not. This p-value will
be the same as the p-value given for testing whether the population slope is 0
or not.
The following Minitab output is for the relationship between pulse rate and
weight in a sample of 35 college women. Notice that .292 is given as the p-value
for testing that the slope is 0 (look under P in the regression results) and for testing that the correlation is 0. Because this is not a small p-value, we cannot reject
the null hypotheses for the slope and the correlation.
Regression Analysis: Pulse Versus Weight
The regression equation is
Pulse = 57.2 + 0.159 Weight
Predictor
Constant
Weight
Coef
57.17
0.1591
SE Coef
18.51
0.1487
T
3.09
1.07
Correlations: Pulse, Weight
Pearson correlation of Pulse and Weight = 0.183
P-Value = 0.292
P
0.004
0.292
Inference About Simple Regression
613
The Effect of Sample Size on Signiﬁcance
14.3 Exercises are on page 627.
technical note
The size of a sample always affects whether a speciﬁc observed result achieves
statistical signiﬁcance. For example, r ϭ 0.183 is not a statistically signiﬁcant
correlation for a sample size of n ϭ 35, as in the pulse and weight example, but
it would be statistically signiﬁcant if n ϭ 1000. With very large sample sizes,
weak relationships with low correlation values can be statistically signiﬁcant.
The “moral of the story” here is that with a large sample size, it may not be saying much to say that two variables are signiﬁcantly related. This means only
that we think that the correlation is not precisely 0. To assess the practical signiﬁcance of the result, we should carefully examine the observed strength of the
relationship.
The usual t-statistic for testing whether the population slope is 0 in a linear regression could also be found by using a formula that involves only n ϭ
sample size and r ϭ correlation between x and y. The algebraic equivalence is
tϭ
b1
r
ϭ 2n Ϫ 2
s.e.1b1 2
21 Ϫ r 2
In the output for the pulse rate and body weight example just given, notice
that the t-statistic for testing whether the slope b1 ϭ 0 is t ϭ 1.07. This was
calculated as
tϭ
b1
0.1591
ϭ
ϭ 1.07
s.e.1b1 2
0.1487
The sample size is n ϭ 35, and the correlation is r ϭ 0.183, so an equivalent
calculation of the t-statistic is
t ϭ 2n Ϫ 2
r
21 Ϫ r
2
ϭ 235 Ϫ 2
0.183
21 Ϫ 0.1832
ϭ 1.07
This second method for calculating the t-statistic illustrates two ideas. First,
there is a direct link between the correlation value and the t-statistic that is
used to test whether the slope is 0. Second, notice that for any ﬁxed value of
r, increasing the sample size n will increase the size of the t-statistic. And the
larger the value of the t-statistic, the stronger is the evidence against the null
hypothesis.
14.4 Predicting y and Estimating
Mean y at a Speciﬁc x
In this section, we cover two different types of intervals that are used to make
inferences about the response variable ( y). The ﬁrst type of interval predicts
the value of y for an individual with a speciﬁc value of x. For example, we may
want to predict the freshman year GPA of a college applicant who has a 3.6 high
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Chapter 14
school GPA. The second type of interval estimates the mean value of y for a population of individuals who all have the same speciﬁc value of x. As an example,
we may want to estimate the mean (average) freshman year GPA of all college
applicants who have a 3.6 high school GPA.
Predicting the Value of y for an Individual
An important use of a regression equation is to estimate or predict the unknown
value of a response variable for an individual with a known speciﬁc value of the
explanatory variable. Using the data described in Example 14.5 (p. 608) for instance, we can predict the maximum distance at which an individual can read
a highway sign by substituting his or her age for x in the sample regression equation. Consider a person who is 21 years old. The predicted distance for such a
person is approximately yˆ ϭ 577 Ϫ 3.01(21) ϭ 513.79, or about 514 feet.
There will be variation among 21-year-olds with regard to the sign-reading
distance, so the predicted distance of 513.79 feet is not likely to be the exact distance for the next 21-year-old who views the sign. Rather than predicting that
the distance will be exactly 513.79 feet, we should instead predict that the distance will be within a particular interval of values.
A 95% prediction interval describes the values of the response variable ( y)
for 95% of all individuals with a particular value of x. This interval can be interpreted in two equivalent ways:
1. The 95% prediction interval estimates the central 95% of the values of y
for members of the population with a speciﬁed value of x.
2. The probability is .95 that a randomly selected individual from the population with a speciﬁed value of x falls into the corresponding 95% prediction interval.
We don’t always have to use a 95% prediction interval. A prediction interval
for the value of the response variable ( y) can be found for any speciﬁed central
percentage of a population with a speciﬁed value of x. For example, a 75% prediction interval describes the central 75% of a population of individuals with a
particular value of the explanatory variable (x).
definition
A prediction interval estimates the value of y for an individual with a particular value of x, or equivalently, the range of values of the response variable for a
speciﬁed central percentage of a population with a particular value of x.
Notice that a prediction interval differs conceptually from a conﬁdence interval. A conﬁdence interval estimates an unknown population parameter,
which is a numerical characteristic or summary of the population. An example
in this chapter is a conﬁdence interval for the slope of the population line. A
prediction interval, however, does not estimate a parameter; instead, it estimates the potential data value for an individual. Equivalently, it describes an interval into which a speciﬁed percentage of the population may fall.
Inference About Simple Regression
615
t h o u g h t q u e s t i o n 1 4 . 4 If we knew the population parameters b0, b1, and s, under the
usual regression assumptions, we would know that the population of y values at a speciﬁc x value was normal with mean b0 ϩ b1x and standard deviation s. In that case,
what interval would cover the central 95% of the y values for that x value? Use your
answer to explain why a prediction interval would not have zero width even with
complete population details.*
As with most regression calculations, the “by hand” formulas for prediction
intervals are formidable. Statistical software can be used to create the interval.
Figure 14.6 shows Minitab output that includes the 95% prediction intervals for
three different ages (21, 30, and 45). The intervals are toward the bottom-right
side of the display in a column labeled “95% PI” and are highlighted with bold
type. The ages for which the intervals were computed are shown at the bottom
of the output. (Note: The term ﬁt is a synonym for yˆ , the estimate of the average
response at the speciﬁc x value.) From Figure 14.6, here is what we can conclude:
●
The probability is .95 that a randomly selected 21-year-old will read the
sign at somewhere between 406.69 and 620.39 feet.
The regression equation is
Distance = 577 – 3.01 Age
Predictor
Constant
Age
S = 49.76
Coef
576.68
–3.0068
R-Sq = 64.2%
SE Coef
23.47
0.4243
T
24.57
–7.09
P
0.000
0.000
R-Sq(adj) = 62.9%
Analysis of Variance
Source
Regression
Residual Error
Total
DF
1
28
29
SS
124333
69334
193667
Unusual Observations
Obs
Age
Distance
27
75.0
460.00
Fit
351.17
MS
124333
2476
F
50.21
SE Fit
13.65
Residual
108.83
P
0.000
St Resid
2.27R
R denotes an observation with a large standardized residual
Predicted Values for New Observations
New Obs
1
2
3
Fit
513.54
486.48
441.37
SE Fit
15.64
12.73
9.44
95.0% CI
(481.50, 545.57)
(460.41, 512.54)
(422.05, 460.70)
95.0% PI
(406.69, 620.39)
(381.26, 591.69)
(337.63, 545.12)
Values of Predictors for New Observations
New Obs
Age
1
21.0
2
30.0
3
45.0
Figure 14.6 ❚ Minitab output showing prediction intervals of distance
*H I N T :
Remember the Empirical Rule, and also recall that the regression equation gives the mean y
for a speciﬁc x.
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Chapter 14
●
The probability is .95 that a randomly selected 30-year-old will read the
sign at somewhere between 381.26 and 591.69 feet.
●
The probability is .95 that a randomly selected 45-year-old will read the
sign at somewhere between 337.63 and 545.12 feet.
We can also interpret each interval as an estimate of the sign-reading distances
for the central 95% of a population of drivers with a speciﬁed age. For instance,
about 95% of all drivers 21 years old will be able to read the sign at a distance
somewhere between roughly 407 and 620 feet.
With Minitab, we can describe any central percentage of the population that
we wish. For example, here are 50% prediction intervals for the sign-reading
distance at the three speciﬁc ages we considered above.
Age
21
30
45
Fit
513.54
486.48
441.37
50.0% PI
(477.89, 549.18)
(451.38, 521.58)
(406.76, 475.98)
For each speciﬁc age, the 50% prediction interval estimates the central 50%
of the maximum sign-reading distances in a population of drivers with that
age. For example, we can estimate that 50% of drivers 21 years old would have
a maximum sign-reading distance somewhere between about 478 feet and
549 feet. The distances for the other 50% of 21-year-old drivers would be predicted to be outside this range, with 25% above about 549 feet and 25% below
about 478 feet.
technical note
The formula for the prediction interval for y at a speciﬁc x is
yˆ Ϯ t *2s 2 ϩ 3 s.e.1ﬁt2 4 2
where
s.e.1ﬁt 2 ϭ s
1x Ϫ x 2 2
1
ϩ
Bn
g 1x1 Ϫ x2 2
The multiplier t * is found by using a t-distribution with n Ϫ 2 degrees of freedom and is such that the probability between Ϫt * and ϩt * equals the desired
level for the interval.
Note:
● The s.e.(ﬁt), and thus the width of the interval, depends on how far the
speciﬁed x value is from x. The farther the speciﬁc x is from the mean, the
wider is the interval.
●
When n is large, s.e.(ﬁt) will be small, and the prediction interval will be
approximately yˆ Ϯ t *s.