Algebra II : Distributions and Curves
Study concepts, example questions & explanations for Algebra II
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Example Questions
Example Question #1 : Normal Distributions
The ages of the students at GW High School are normally distributed with a mean of 
What is the proportion of students that are younger than 
Not enough information to answer the question.
This question relates to the 
In this case this means that 
Therefore we have 
Thus the right answer is 
Example Question #1 : Normal Distributions
The scores for your recent english test follow a normal distribution pattern. The mean was a 75 and the standard deviation was 4 points. What percentage of the scores were below a 67?
7.5%
5%
10%
2.5%
2.5%
Use the 68-95-99.7 rule which states that 68% of the data is within 1 standard deviation (in either direction) of the mean, 95% is within 2 standard deviations, and 99.7% is within 3 standard deviations of the mean.
In this case, 95% of the students' scores were between:
75-(2 x 4) and 75+(2 x 4)
or between a 67 and a 83, with equal amounts of the leftover 5% of scores above and below those scores. This would mean that 2.5% of the students scored below a 67% on the test.
Example Question #491 : Algebra Ii
Your class just took a math test. The mean test score was a 78 with a standard deviation of 2 points. With this being the case, 99.7% of the class scored between what two scores?
Use the 68-95-99.7 rule which states that 68% of the data is within 1 standard deviation (in either direction) of the mean, 95% is within 2 standard deviations, and 99.7% is within 3 standard deviations (in either direction) of the mean.
In this case, 99.7% of the students' scores were between 3 standard deviation above the mean and 3 standard deviations below the mean:
78-(2 x 3) and 78+(2 x 3)
or between a 72 and an 84.
Example Question #2 : Graphing Data
All of the following statements regarding a Normal Distribution are true except:
All of these are true.
A graph of a normally-distributed data set will have a single, central peak at the mean of the data set that it describes.
Between two graphs of normally-distributed data sets, the graph of the set with a higher standard deviation will be wider than the graph of the set with a lower standard deviation.
A graph of a normally-distributed data set is symmetrical.
The shape of the graph of a normally-distributed data set is dependent upon the mean and the standard deviation of the data set that it describes.
All of these are true.
The graph of a normally-distributed data set is symmetrical.
The graph of a normally-distributed data set has a single, central peak at the mean of the data set that it describes.
The graph of a normally-distributed data set will vary based only upon the mean and the standard deviation of the set that it describes.
The graph of a normally-distributed data set with a higher standard deviation will be wider than the graph of a normally-distributed data set with a lower standard deviation.
The question asks us to find the statement that is not true; however, all statements are true so the correct response is "All of these are true."
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 
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