Algebra II : Z-scores
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Example Questions
Example Question #1 : Z Scores
A large group of test scores is normally distributed with mean 78.2 and standard deviation 4.3. What percent of the students scored 85 or better (nearest whole percent)?
If the mean of a normally distributed set of scores is 
From a 
We want the percent of students whose test score is 85 or better, so we want 
or about 5.7 % The correct choice is 6%.
Example Question #2 : Z Scores
The salaries of employees at XYZ Corporation follow a normal distribution with mean 60,000 and standard deviation 7,500. What proportion of employees earn approximately between 69,000 and 78,000?
Use the normal distribution table to calculate the probabilities. Round your answer to the nearest thousandth.
Let X represent the salaries of employees at XYZ Corporation.
We want to determine the probability that X is between 69,000 and 78,000:
To approximate this probability, we convert 69,000 and 78,000 to standardized values (z-scores).
We then want to determine the probability that z is between 1.2 and 2.4
The proportion of employees who earn between 69,000 and 78,000 is 0.107.
Example Question #2 : Z Scores
On a statistics exam, the mean score was 
The z-score is a measure of an actual score's distance from the mean in terms of the standard deviation. The formula is:
Where 
If we plug in the values we have from the original problem we have
which is approximately 
Example Question #1 : Graphing Data
A distributor manufactures a product that has an average weight of 
If the standard deviation is 
The z-score can be expressed as
where
Therefore the z-score is:
Example Question #1 : Z Scores
The mean grade on a science test was 79 and there was a standard deviation of 6. If your sister scored an 88, what is her z-score?
Use the formula for z-score:
Where 
Example Question #1 : Graphing Data
Your teacher tells you that the mean score for a test was a 
You are given that the 
The formula for a z-score is
where 
Plugging in your z-score, mean, and standard deviation that was originally given in the question we get the following.
Now to find the grade you got on the test we will solve for 
Example Question #1 : Z Scores
In a normal distribution, if the mean score is 8 in a gymnastics competition and the student scores a 9.3, what is the z-score if the standard deviation is 2.5?
Write the formula to find the z-score. Z-scores are defined as the number of standard deviations from the given mean.
Substitute the values into the formula and solve for the z-score.
Example Question #1 : Z Scores
Suppose a student scored a 
Write the formula for z-score where 
Substitute the variables.
The z-score is: 
Example Question #2 : Z Scores
Suppose Bob's test score is 50. Determine the z-score if the standard deviation is 3, and the mean is 75.
Write the formula for z-scores. This tells how many standard deviations above the below the mean.
Substitute the known values into the equation.
The z-score is:
Example Question #2 : Z Scores
Find the z-score if the result of a test score is 6, the mean is 8, the standard deviation is 2.
Write the formula to determine the z-scores.
Substitute all the known values into the formula to determine the z-score.
Simplify this equation.
The answer is:
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