Answer :
Final answer:
The salaries $64,176 and $44,176 correspond to one standard deviation above and below the mean respectively. $50,000 is less than one standard deviation below the mean, and $74,176 is almost two standard deviations above the mean.
Explanation:
In this question, we are dealing with the concept of standard deviation - a measure of how spread out numbers are from the mean. Given that the average teacher's salary is $54,176 and the standard deviation is $10,430:
- A salary of $64,176 is one standard deviation above the mean because $64,176 - $54,176 = $10,430, which is exactly the standard deviation.
- A salary of $44,176 is one standard deviation below the mean because $54,176 - $44,176 = $10,430, again the exact standard deviation.
- A salary of $50,000 is less than one standard deviation below the mean because $54,176 - $50,000 = $4,176, which is less than the standard deviation of $10,430.
- A salary of $74,176 is almost two standard deviations above the mean because $74,176 - $54,176 = $20,000, which is close to twice the standard deviation of $10,430.
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Final answer:
To find the salaries corresponding to given scores, use z-scores and the formula: Salary = Mean + (z-score x Standard Deviation)
Explanation:
To find the salaries corresponding to the given scores, we need to use z-scores and the formula:
Salary = Mean + (z-score x Standard Deviation)
a) $64,176:
z-score = (64,176 - 54,176) / 10,430 = 0.9589
Salary = 54,176 + (0.9589 x 10,430) = $64,446.50
b) $44,176:
z-score = (44,176 - 54,176) / 10,430 = -0.9589
Salary = 54,176 + (-0.9589 x 10,430) = $43,905.50
c) $50,000:
z-score = (50,000 - 54,176) / 10,430 = -0.3995
Salary = 54,176 + (-0.3995 x 10,430) = $49,172.80
d) $74,176:
z-score = (74,176 - 54,176) / 10,430 = 1.9176
Salary = 54,176 + (1.9176 x 10,430) = $74,196.50
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