Suppose a local university researcher wants to build a linear model that predicts the freshman year GPA of incoming students based on high school SAT scores. The researcher randomly selects a sample of 40 sophomore students at the university and gathers their freshman year GPA data and the high school SAT score reported on each of their college applications. He produces a scatterplot with SAT scores on the horizontal axis and GPA on the vertical axis. The data has a linear correlation coefficient of 0.503202. Additional sample statistics are summarized in the table below. Variable Variable description Sample mean Sample standard deviation ?x high school SAT score ?⎯⎯⎯=x¯= 1501.717708 ??=104.141305sx=104.141305 ?y freshman year GPA ?⎯⎯⎯=3.300318y¯=3.300318 ??=0.451901sy=0.451901 ?slope=0.503202=0.002184r=0.503202slope=0.002184 Determine the ?y‑intercept, ?a, of the least-squares regression line for this data. Give your answer precise to
Inverse Normal Distribution
The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.
Mean, Median, Mode
It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.
Z-Scores
A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.
Suppose a local university researcher wants to build a linear model that predicts the freshman year GPA of incoming students based on high school SAT scores. The researcher randomly selects a sample of 40 sophomore students at the university and gathers their freshman year GPA data and the high school SAT score reported on each of their college applications. He produces a
| Variable | Variable description |
Sample mean |
Sample standard deviation |
|---|---|---|---|
| ?x | high school SAT score | ?⎯⎯⎯=x¯= 1501.717708 | ??=104.141305sx=104.141305 |
| ?y | freshman year GPA | ?⎯⎯⎯=3.300318y¯=3.300318 | ??=0.451901sy=0.451901 |
Determine the ?y‑intercept, ?a, of the least-squares regression line for this data. Give your answer precise to at least four decimal places.
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