AP Statistics – Chapter 3 [Sections 1 and 2] Review of Selected Exercises:
3.21:
a)Median household income is about $32,800.
Mean income per person is about $27,500.
[These solutions were found just by looking at the graph provided by the situation]
b)I expect a positive association between these variables because both measure the prosperity of a state; if one measure is high, chances are that the other will also be high.
Household income is expected to be higher because most households have multiple sources of income.
c)The mean income per person in a state can be higher than the median household income if there are some individuals earning significantly more money than the rest of the distribution [mean is not resistant to outliers].
d)Alaska’s median household income is about $48,000
e)Ignoring the outliers, the distribution is positive in direction, linear in form, and moderately strong.
3.22:
a)The explanatory variable in this situation is time.
The response variable is pulse.
b)There is a negative association. A higher speed [lower time] will correspond with a higher pulse and slower speeds [higher time] will correspond with lower pulses.
c)This negative, linear relationship is moderately strong.
d)r = -0.744. This value seems reasonable because there is an obvious negative correlation, but the values are not tightly packed together [small times correspond with large pulses and large times correspond with small pulses]
e)If the times had been recorded in seconds, the value of r would not change. Since r uses standardized values of the observations, r does not change when we change the units of measurement of x, y, or both.
3.23:
a)Gender is a categorical variable and r measures the strength of a linear relationship between quantitative variables only.
b)The correlation r cannot be larger than 1.
c)The correlation r has no units!
3.24:
The psychologist meant that there is no linear relationship between research productivity and teaching rating.
When r = 0, the only thing we can state for sure is that there is no linear relationship between the variables.
Remember: Correlation r describes the strength of a linear relationship only!
3.27:
r = 0.4811.
The correlation is lowered because of the outlier (10, 1).
The data, minus the outlier, shows a strong linear relationship.
If the outlier was eliminated, the r value would be closer to 1.
3.49:
a)The slope tells us that on average, BOD rises by 1.507 mg/L for every 1 mg/L increase in TOC.
b)When TOC = 0 mh/L, the predicted BOD level is -55.43 mh/L.
The negative value of BOD was obtained because values of TOC near zero were probably not included in the study. This model would not be effective for TOC values near zero [with TOC values near zero, the BOD values are negative, which is impossible].
3.50:
a)
x = IQ [explanatory variable]
y = GPA [response variable]
b) Therefore, 40.16% of the variation in GPA is explained by the least-squares regression line with IQ.
c)Predicted GPA = 6.8511 []
The residual for this student is - 6.3211 [since their actual GPA was 0.53 (this verifies that the actual value is below the regression line)]
3.53:
a)Scatterplot shows a strong linear pattern.
b)
x = Age [explanatory variable]
y = Height [response variable]
c) and
Predicted Height at 40 months = 87.28 cm
Predicted Height at 60 months = 94.95 cm
- Note: To draw the linear regression line on your calculator:
- STAT CALC
- LinReg(a+bx)L1, L2, Y1
- Y1 is found under VARS Y-VARS 1: Function
- ZOOM 9: ZoomStat
3.54:
a)
Predicted Height at 40 years (480 months) = 255.93 cm [100.7596 inches]
This is just about 8.4 feet!
b)This prediction is impossible large, due to the fact that extrapolation was used.
Remember that extrapolation often produces unreliable predictions.
3.55:
a)b = 0.16
For every point earned on the midterm, the score on the final exam is predicted to increase by 0.16.
a = 30.2
If the student had a pre-exam total of 0 points, the predicted score on the final exam would be 30.2.
b)Using the regression line, Julie’s predicted final exam score is 78.2
c)
This means that only 36% of the variability in final exam scores is explained by the least squares regression line with pre-exam totals.
Therefore, Julie has reason to believe this is not a good estimate.
3.58:
This regression line explains the connection between midterm exam grades and final exam grades.
This student scored 10 points higher than the mean on the midterm [].
This student’s predicted final exam grade can be found by inputting his midterm grade into the equation.
This student’s predicted final exam score is 4.1 points above the class mean.
This is an example of the phenomenon that gave “regression” its name:
[Students who do well on the midterm will on the average to less well, but still above average, on the final]