Distributions and Curves - Algebra 2
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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?
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?
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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.
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.
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?
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?
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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.
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.
The ages of the students at GW High School are normally distributed with a mean of 
and a standard deviation of 
years.
What is the proportion of students that are younger than 
years old?
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
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This question relates to the 
rule of normal distribution. We know that 
of the data are within 
standard deviations from the mean.
In this case this means that 
of the students are between
and 
and 
and 
Therefore we have 
of the students that are outside of this range. Since the normal distribution is symmetric, the proportion of students below 
is the same as the proportion of students above 
.
Thus the right answer is 
or 
.
This question relates to the
In this case this means that
Therefore we have
Thus the right answer is
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?
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?
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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.
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.
All of the following statements regarding a Normal Distribution are true except:
All of the following statements regarding a Normal Distribution are true except:
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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."
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."
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)?
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)?
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If the mean of a normally distributed set of scores is 
and the standard deviation is 
, then the 
-score corresponding to a test score of 
is
From a 
-score table, in a normal distribution,
We want the percent of students whose test score is 85 or better, so we want 
. This is
or about 5.7 % The correct choice is 6%.
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%.
A distributor manufactures a product that has an average weight of 
pounds.
If the standard deviation is 
pounds, determine the z-score of a product that has a weight of 
pounds.
A distributor manufactures a product that has an average weight of
If the standard deviation is
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The z-score can be expressed as
where
Therefore the z-score is:
The z-score can be expressed as
where
Therefore the z-score is:
On a statistics exam, the mean score was 
and there was a standard deviation of 
. If a student's actual score of 
, what is his/her z-score?
On a statistics exam, the mean score was
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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 
are the mean and standard deviation, respectively. 
is the actual score.
If we plug in the values we have from the original problem we have
which is approximately 
.
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
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.
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.
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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.
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.
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?
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?
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Use the formula for z-score:
Where 
is her test score, 
is the mean, and 
is the standard deviation.
Use the formula for z-score:
Where
You just took your ACT. The mean score was a 
with a standard deviation of 
. If you scored a 
, what is your z-score?
You just took your ACT. The mean score was a 
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Use the formula for z-score:
Where 
is your score, 
is the mean, and 
is the standard deviation.
Use the formula for z-score:
Where
Your teacher tells you that the mean score for a test was a 
and that the standard deviation was 
for your class.
You are given that the 
-score for your test was 
. What did you score on your test?
Your teacher tells you that the mean score for a test was a
You are given that the
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The formula for a z-score is
where 
= mean and 
= standard deviation and 
=your test grade.
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 
.
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
Suppose a student scored a 
on a test. The mean of the tests are 
, and the standard deviation is 
. What is the student's z-score?
Suppose a student scored a
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Write the formula for z-score where 
is the data, 
is the population mean, and 
is the population standard deviation.
Substitute the variables.
The z-score is: 
Write the formula for z-score where
Substitute the variables.
The z-score is:
Suppose Bob's test score is 50. Determine the z-score if the standard deviation is 3, and the mean is 75.
Suppose Bob's test score is 50. Determine the z-score if the standard deviation is 3, and the mean is 75.
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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: 
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:
Find the z-score if the result of a test score is 6, the mean is 8, the standard deviation is 2.
Find the z-score if the result of a test score is 6, the mean is 8, the standard deviation is 2.
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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: 
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:
Determine the z-score of a test result is 45, the mean is 60, and the standard deviation is 8.
Determine the z-score of a test result is 45, the mean is 60, and the standard deviation is 8.
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Write the formula for z-scores. Z-scores determine the number of standard deviations below or above the mean.
Substitute the values into the formula to determine the z-score.
The z-score is 
.
Write the formula for z-scores. Z-scores determine the number of standard deviations below or above the mean.
Substitute the values into the formula to determine the z-score.
The z-score is
Suppose Billy scored a 54 on his exam. The mean of the exam grades is 66 and the standard deviation is 3. What is the z-score?
Suppose Billy scored a 54 on his exam. The mean of the exam grades is 66 and the standard deviation is 3. What is the z-score?
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Write the formula for z-scores. Z scores determine how many standard deviations a score is above or below the mean.
Substitute the known values in order to determine the z-score.
The answer is: 
Write the formula for z-scores. Z scores determine how many standard deviations a score is above or below the mean.
Substitute the known values in order to determine the z-score.
The answer is:
Sarah scored an 8.5 out of ten on her gymnastics floor routine. If the mean of the scores is 9.2 and the standard deviation is 1.3, what is her z-score?
Sarah scored an 8.5 out of ten on her gymnastics floor routine. If the mean of the scores is 9.2 and the standard deviation is 1.3, what is her z-score?
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Write the formula for z-scores. Z-scores are indicators of how many standard deviations above or below the mean.
Substitute the known values.
The answer is: 
Write the formula for z-scores. Z-scores are indicators of how many standard deviations above or below the mean.
Substitute the known values.
The answer is:
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?
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?
Tap to see back →
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.
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.
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?
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?
Tap to see back →
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.
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.