In a study of the legibility and visibility of highway signs, a Pennsylvania res
ID: 3066018 • Letter: I
Question
In a study of the legibility and visibility of highway signs, a Pennsylvania research firm determined the maximum distance (in feet) at which each of 30 drivers could read a newly designed sign. The 30 participants were volunteers and ranged in age from 18 to 82 years old. The government agency that funded the research hoped to improve highway safety for older drivers and wanted to examine the relationship between age and the sign legibility distance. A scatterplot of the data is shown below. The sample correlation coefficient is r=0.801r=0.801 and the least-squares regression line is
y =576.683.01xy^=576.683.01x
Information
Driver age (years) and the maximum distance at which a highway sign was read (feet) for a volunteer sample of 30 drivers.
a). One of the study participant's 5-year old daughter could read the highway sign at a maximum distance of 530 feet. The residual for this 5-year-old is
zero.
negative.
positive.
cannot be determined with the information given.
b). Is the predicted maximum distance at which a 5-year-old can read the highway sign a valid prediction?
No, since these data only show correlation and not a cause-and-effect relationship.
Yes, since the regression line is valid for all ages.
No, since this is an example of extrapolation.
Yes, since the scatterplot shows a linear relationship.
c). What are the appropriate null and alternative hypotheses for a hypothesis test of the population correlation coefficient in notation?
H0:=0Ha:0H0:=0Ha:0
H0:b=0Ha:b0H0:b=0Ha:b0
H0:=0Ha:0H0:=0Ha:0
H0:r=0Ha:r0H0:r=0Ha:r0
d). A null distribution of 5000 simulated sample correlation coefficients is shown below. How would you use this distribution to calculate a p-value for these data?
Calculate the proportion of simulated correlation coefficients that are beyond 576.68.
Calculate the proportion of simulated correlation coefficients that are beyond -0.801.
Calculate the proportion of simulated correlation coefficients that are beyond 0.
Calculate the proportion of simulated correlation coefficients that are beyond -3.01.
e). How was one sample generated in the null distribution shown in the previous question?
Flip a coin for each pair; if heads, swap the x and y values, otherwise, do not change the values.
Shift the values so that the correlation coefficient is zero, then sample with replacement 30 times from the shifted data.
Hold the x-values constant and re-randomize the y-values to the x-values.
Label cards with the y-values, mix them together, and shuffle them into two groups.
zero.
negative.
positive.
cannot be determined with the information given.
Explanation / Answer
a) predicted value =576.683.01*5=561.63
as actual value is less than predicted value ; residual is negative
b)No, since this is an example of extrapolation
c)=0Ha:0
d)Calculate the proportion of simulated correlation coefficients that are beyond -0.801.
e)
Hold the x-values constant and re-randomize the y-values to the x-values.
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