Algebra II : Distributions and Curves

Study concepts, example questions & explanations for Algebra II

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Example Questions

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Example Question #1 : Graphing Data

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?

Possible Answers:

Not enough information to answer the question.

Correct answer:

Explanation:

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 .

 

Example Question #491 : Algebra Ii

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?

Possible Answers:

5%

2.5%

7.5%

10%

Correct answer:

2.5%

Explanation:

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 #1 : Normal Distributions

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?

Possible Answers:

Correct answer:

Explanation:

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 #1 : Normal Distributions

All of the following statements regarding a Normal Distribution are true except:

Possible Answers:

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.

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.

Correct answer:

All of these are true.

Explanation:

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)?

Possible Answers:

Correct answer:

Explanation:

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%.

Example Question #1 : 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?

 

Normal-distribution

Use the normal distribution table to calculate the probabilities. Round your answer to the nearest thousandth. 

Possible Answers:

Correct answer:

Explanation:

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 #3 : Z Scores

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?

Possible Answers:

Correct answer:

Explanation:

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 .

 

Example Question #1 : Z Scores

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.

Possible Answers:

Correct answer:

Explanation:

The z-score can be expressed as

where 

Therefore the z-score is:

Example Question #5 : 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?

Possible Answers:

Correct answer:

Explanation:

Use the formula for z-score:

Where  is her test score,  is the mean, and  is the standard deviation.

Example Question #6 : Z Scores

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?

Possible Answers:

Correct answer:

Explanation:

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 .

 

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