Gate count—Computing the least squares line from descriptive statistics. Many li
ID: 2932851 • Letter: G
Question
Gate count—Computing the least squares line from descriptive statistics. Many libraries have gates that automatically count persons as they leave the building, thus making it easy to find out how many persons used the library in a given year. (Of course, someone who enters, leaves, and enters again is counted twice.) Researchers conducted a survey of liberal arts college libraries and found the following descriptive statistics on enrollment and gate count:
N MEAN MEDIAN TRMEAN STDEV SEMEAN 159 Enroll 17 20092007 2024 657 Gate 17 247235 254116 247827 10480725419 Correlation of Enroll and Gate = 0.701 The least squares regression line can be computed from five descriptive statistics, as follows: where T, 7, Sr, and sy are the sample means and standard deviations for the predictor and response, respectively, and r is their correlation. a. Find the equation of the least squares line for predicting the gate count from enrollment b. What percentage of the variation in the gate counts is explained by enrollments? c. Predict the number of persons who will use the library at a small liberal arts college with an enrollment of 1445. d. One of the reporting colleges has an enrollment of 2200 and a gate count of 130,000. Find the value of the residual for this college.Explanation / Answer
a . Equation of linear regression line
y^ = 0 + 1 x
1 = r * (sy/sx) = 0.701 * (104807/657) = 111.826
as the mean values of x and y will pass through the regression line so putting this value in regression line, we get
x = 2009 , = 247235
y^ = 0 + 1 x
= 0 + 1x
247235 = 0 + 111.826 * 2009
0 = 22572.55
y^ = 22572.55 + 111.826x
Gate count = 22572.55 + 111.826 * enrollment.
(ii) Here r = 0.701
R2 = 0.4914
so here we can say that 49.14% variation in the gate counts can be explained by number of enrolments.
(c) If enrolment = 1445
Gate Count = 22572.55 + 111.826 * 1445 = 184161 gate counts
(d) Enrollment = 2200
predicted Gate count = 22572.55 + 111.826 * 2200 = 268590
The real gate count = 130,000
so residual for this college = 130000 - 268590 = -138590
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