Chapter 14 Write-ups
Problem 10.
Step 1: Linear regression t-interval
x = speed
y = steps per second
Step 2:
There is a strong positive linear relationship
between speed and steps per second of competitive
runners.
By linear regression: ŷ = 1.76608 + 0.080284x
The residual plot has a strong pattern,
bringing into question the linear model.
Examination of the magnitudes of these
residuals, however, shows that they are
quite small, ranging only from -0.01 to
0.01, so we may be able to tolerate that
amount of pattern.
The standard deviation is relatively constant throughout.
The data appear to be independent, and normal procedures should guarantee that.
The normal quantile plot (normal probability plot) of
the residuals is roughly linear, suggesting a normal
model for the residuals.
Chapter 14 Write-ups
Step 3: t =
Step 3:
b
SEb
b ± t * SEb
0.080284 ± 4.032(0.0016)
(0.0738,0.0867)
Step 4: We are 99% confident that the slope of the linear relationship between speed and number
of steps per second falls within (0.0738, 0.0867).
Now for a test of significance:
Step 1: H 0 : b = 0
Ha : b > 0
b is the slope of the line between running speed and number of steps per second for competitive
runners.
Step 2: Same as above.
b
0.080284
=
= 49.7 df = 5
SEb
0.0016
Step 4: The shaded region is too small to see.
Step 3: t =
Step 5: P(t > 49.7) = 3.12 ´10-8
Step 6: Reject H0, a test statistic this large
-8
by chance alone. p = 3.12 ´10 < .01 = a
will rarely happen
Step 7: We have strong evidence of a positive linear relationship between speed and number of
steps for competitive runners.
Problem 11
Step 1: Linear regression t-interval
x = year
y = lean measured by tenths of mm over 2.9 m
Step 2:
There is a strong positive linear relationship between
year and amount of lean of the Leaning Tower of Pisa.
Linear regression produces:
ŷ = -61.1209 + 9.31868x
Chapter 14 Write-ups
The residual plot is patternless, confirming
the linear model. It also shows that the
standard deviation remains about the same
throughout.
The normal normal probability plot of the residuals is fairly
nonlinear, calling into question whether the residuals vary
normally.
This data collected over time must give independent measures.
Step 3: b ± t * SEb
df = 11
9.31868 ± 2.201(0.3099)
(8.636,10.000)
Step 4: We are 95% confident that the slope of the linear relationship between year and lean of
the Tower of Pisa falls within (8.636, 10.000). (There was a problem meeting the requirement
that the residuals vary normally.)
Now for a test of significance:
Step 1: H 0 : b = 0
Ha : b > 0
b is the slope of the line between the year and the amount of lean of the Tower of Pisa.
Step 2: Same as above.
Step 3: t =
Step 4:
b
9.31868
=
= 30.1 df = 11
SEb 0.3099
The shaded region is too small to see.
Step 5: P(t > 30.1) < 0.0001
Step 6: Reject H0, a test statistic this large will rarely
-4
happen by chance alone. p < 10 < .01 = a
Step 7: We have strong evidence of a positive linear relationship between the year and the
amount of lean of the Tower of Pisa. (There was a problem meeting the requirement that the
residuals vary normally.)
Chapter 14 Write-ups
Problem 15
Step 1: Linear regression t interval for slope
Slope =
D metabolic rate
D mass
Step 2:
This scatterplot of mass vs. metabolic rate shows a
strong positive linear relationship. The least squares
line is ŷ = 113.165 + 26.878x where x is mass in kg
and y in calories.
The residual plot of mass vs. residuals is
patternless, confirming the linear model.
standard deviation appears to be
approximately the same throughout.
The
The normal probability plot of the residuals is only
very roughly linear, so there is uncertainty about
normality, giving a possible assumption violation.
2.6
410
-207
Ordinary experimental procedures will give
independent data, so this assumption appears to be
met.
-2.6
b ± t * SEb
26.878 ± 1.740(3.786)
(20.29, 33.46)
df = 17
We are 90% confident that the true slope of the regression line between mass and metabolic rate
is between 20.9 and 33.4 calories per kg. There was an assumption violation, however. In
repeated random sampling this method captures the true slope 90% of the time.