Concept explainers
(a)
The linear transformation that change third-grade scores x into new scores
(a)
Answer to Problem 176E
Solution: The linear transformation that change third-grade scores x into new scores is
Explanation of Solution
Given: Each grade average of 100 and standard deviation is 20.
Calculation: The reading ability has mean 70 and standard deviation 10 provided to third-graders and has mean 80 and standard deviation 11 on the same test and the required new marks are
The new score transformation is given by
Now, the mean of the new marks will be:
That is,
And the standard deviation of new score is:
For third graders standard deviation is
Substitute the values in the equation (2) and get,
For third graders mean is
Interpretation: The linear transformation that change third-grade scores x into new scores is
(b)
The linear transformation that change sixth-grade scores x into new scores
(b)
Answer to Problem 176E
Solution: The linear transformation that change sixth-grade scores x into new scores is
Explanation of Solution
Given: The reading ability has mean 70 and standard deviation 10 provided to sixth-graders and has mean 80 and standard deviation 11 on the same test.
The third-grade marks are
That is,
And the standard deviation of new score is:
For sixth-graders standard deviation is
For sixth-graders mean is
Interpretation: The linear transformation that change sixth-grade scores x into new scores is
(c)
To find: The transformed score of David and Nancy if David is a third-grade student and scores 72 in the test and Nancy is a sixth-grade student and scores 78 in the test and to find who scores more in the test after transformation.
(c)
Answer to Problem 176E
Solution: The transformed score of David is 104 and the transformed score of Nancy is 96.3638. David scores more than Nancy after transformation into new scores.
Explanation of Solution
Given: The linear transformation that change third-grade scores x into new scores is
Calculation: David scores 72 in the test with the third-grade. So, his transformed new score can be calculated as:
This score is above the mean of 100. And Nancy scores 72 in the test with the third-grade. So, her transformed new score can be calculated as:
This score is below the mean of 100.
Interpretation: Therefore, David scores more than Nancy after transformation into new scores.
(d)
To find: What percent of third-graders and sixth-graders have scored less than 75 if the distribution of scores in both grade is normal and have
(d)
Answer to Problem 176E
Solution: Around 10.56% of third-graders and sixth-graders have scored less than 75.
Explanation of Solution
Calculation: The Z score is given by the formula,
Substitute the values in the formula for standardized score of 75 as:
From the standard normal table
Interpretation: Therefore, it can be concluded that 10.56% of third-graders and sixth-graders have scored less than 75.
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Chapter 1 Solutions
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