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Algebra II : Algebra II

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #1 : Percentiles

Stem_and_leaf

Above is the stem-and-leaf display for a group of test scores. Which of the following scores would come closest to being at the $35^{th}$ percentile?

Possible Answers:

Correct answer:

Explanation:

The stem-and-leaf display represents 53 scores. The score at the 35th percentile would be the score that is greater than 35% of the scores, or

$53 \times 0.35 \approx 18.5$ scores.

Rounding, we count 19 scores from the bottom:

Untitled

The score represented is 64, which is the correct response.

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Example Question #1 : Deviation Concepts

Calculate the variance of the following set of values.

$\left \{ 3,3,4,5,8,7\right \}$

Possible Answers:

3.67

1.91

1.67

Correct answer:

3.67

Explanation:

Variance is the average of the squared differences from the mean. So start by working out the mean of the set of values.

$\frac{(3+3+4+5+8+7)}{6}=5$

Then, for each number, subtract the mean and square the result.

$(3-5)^{2}=(-2)^{2}=4$

$(4-5)^{2}=(-1)^{2}=1$

$(5-5)^{2}=(0)^{2}=0$

$(8-5)^{2}=(3)^{2}=9$

$(7-5)^{2}=(2)^{2}=4$

The average of the results is the variance.

$\frac{(4+4+1+0+9+4)}{6}=3.67$

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Example Question #1 : Variance

What is the variance of the following set of numbers?

(17,13,21,14,10)

Possible Answers:

$5\sqrt{3}$

$2\sqrt{6}$

Correct answer:

Explanation:

Remember that the variance of a set of numbers is equal to its standard deviation squared. So if the formula for standard deviation is known to be:

$\sigma =\sqrt{\frac{1}{N}\sum_{i=1}^{N}(x_i-\mu )^2}$

Then the variance is simply $\sigma^2$ , given by the following equation:

$\sigma^2 =\frac{1}{N}\sum_{i=1}^{N}(x_i-\mu )^2$

Where $\sigma^2$ is the variance, N is the number of values in the set, x_i is the number currently being evaluated in the summation, and $\mu$ is the mean of the set. So all we must do to find the variance is determine our mean, and then sum the values given by subtracting the mean from each number in the set and squaring it, dividing the final sum by N to give us our variance. We start by determining the mean of the set:

$\mu=\frac{\sum_{i=1}^{N}x_i}{N}=\frac{17+13+21+14+10}{5}=15$

We then find each value that will be added together by subtracting our mean from each of the numbers in our set and then squaring the result:

(17-15)^2=4

(13-15)^2=4

(21-15)^2=36

(14-15)^2=1

(10-15)^2=25

Looking at our formula for variance, all we must do now is add these values together and divide by N:

$\sigma^2 =\frac{1}{5}(4+4+36+1+25)=14$

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Example Question #2 : Deviation Concepts

A set of five numbers is given as follows, where one of the numbers, , is unknown:

(37,42,39,33,x)

If the mean of the set is , what is the variance?

Possible Answers:

35.4

24.7

28.8

31.6

19.9

Correct answer:

28.8

Explanation:

Before we can determine the variance of the set, we must first determine the value of x, our unknown. The problem tells us the mean of the set, and we know each number in the set except for x, so we can solve for x as follows:

$\mu =\frac{\sum_{i=1}^{N}x_i}{N}=\frac{37+42+39+33+x}{5}=40$

$151+x=200\rightarrow x=49$

Now that we know all the values of the set, the mean, and the number of values, we can solve for the variance using the following equation:

$\sigma ^2=\frac{1}{N}\sum_{i=1}^{N}(x_i-\mu )^2$

Where $\sigma ^2$ is the variance, N is the number of values in the set, x_i is the value currently being evaluated in the set, and $\mu$ is the mean of the set. We start by determing the value of each term in the summation part of the formula:

(37-40)^2=9

(42-40)^2=4

(39-40)^2=1

(33-40)^2=49

(49-40)^2=81

Now that we now each value of the summation, all we have left to do is add them all together and divide the final result by N:

$\sigma ^2=\frac{1}{5}(9+4+1+49+81)=\frac{144}{5}=28.8$

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Example Question #4 : Variance

What is the variance for the values , , and ?

Possible Answers:

$\frac{2}{9}$

$\frac{2}{3}$

$\frac{4}{5}$

$\frac{3}{2}$

Correct answer:

$\frac{2}{3}$

Explanation:

Write the formula for variance.

$\sigma^2 = \frac{\sum (X-\mu)^2 }{N}$

The value of represents each number in the data set.

The value of is 3, since there are 3 numbers in the data set.

Find the mean, $\mu$ .

$\mu= \frac{2+3+4}{3} = \frac{9}{3}=3$

Rewrite the variance formula:

$\sigma^2 = \frac{(2-3)^2 +(3-3)^2+(4-3)^2}{3} = \frac{1+0+1}{3}=\frac{2}{3}$

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Example Question #31 : Statistical Concepts

Determine the variance if the standard deviation is 8.6 .

Possible Answers:

73.70

73.96

25.22

2.15

2.93

Correct answer:

73.96

Explanation:

Write the formula relationship between standard deviation and variance.

The standard deviation is the square root of variance.

$\sigma = \sqrt{\textup{Variance}}$

Substitute the standard deviation.

$8.6 = \sqrt{\textup{Variance}}$

Square both sides.

$\textup{Variance} =( 8.6)^2 = 73.96$

The answer is: 73.96

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Example Question #6 : Variance

Evaluate the variance if the standard deviation is 19.5 .

Possible Answers:

7414.875

380.25

1.29

4.416

2.692

Correct answer:

380.25

Explanation:

Write the formula for the variance.

$\textup{Variance} =\sigma^2$

$\textup{Standard Deviation}= \sigma$

The standard deviation is the square root of variance.

$\sigma = \sqrt{\textup{Variance}}$

Substitute the value of standard deviation in the formula.

$19.5 = \sqrt{\textup{Variance}}$

Square both sides.

$\textup{Variance }= 380.25$

The answer is: 380.25

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Example Question #1 : Deviation Concepts

The standard deviation of a distribution is 9. What is the variance?

Possible Answers:

Correct answer:

Explanation:

The variance of a distribution is equal to the standard deviation squared.

9^2=81

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Example Question #8 : Variance

Determine the population variance given the following data set: [5,7,27]

Possible Answers:

$\frac{296}{3}$

148

280

Correct answer:

$\frac{296}{3}$

Explanation:

To determine the variance, we will need to find the squared differences of the mean, and take the average.

Find the mean.

$\frac{5+7+27}{3} = \frac{39}{3}=13$

Subtract each number from the mean and square the quantities.

(5-13)^2+(7-13)^2+(27-13)^2 = 64+36+196 = 296

Divide this value by three.

The answer is: $\frac{296}{3}$

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Example Question #1 : Standard Deviation

Calculate the standard deviation of the following set of values.

$\left \{ 2,3,3,4,5,7\right \}$

Possible Answers:

$\frac{4}{3}$

$\frac{8}{3}$

Correct answer:

Explanation:

To get the standard deviation, we need to calculate the variance, which is the average of the squared differences from the mean, so we will start by getting the mean.

$\frac{(2+3+3+4+5+7)}{6}=4$

Then, subtract the mean from each value...

$(2-4)^{2}=(-2)^{2}=4$

$(3-4)^{2}=(-1)^{2}=1$

$(4-4)^{2}=(0)^{2}=0$

$(5-4)^{2}=(1)^{2}=1$

$(7-4)^{2}=(3)^{2}=9$

...and take the mean of these resulting values, which is equal to the variance.

$\frac{(4+1+1+0+1+9)}{6}=2\frac{2}{3}=\frac{8}{3}$

The square root of this value is the standard deviation. The answer is presented as $\sqrt{\frac{8}{3}}$ ,

but you may also calculate it and find it equal to about 1.63

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Algebra II : Algebra II

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #1 : Percentiles

Example Question #1 : Deviation Concepts

Example Question #1 : Variance

Example Question #2 : Deviation Concepts

Example Question #4 : Variance

Example Question #31 : Statistical Concepts

Example Question #6 : Variance

Example Question #1 : Deviation Concepts

Example Question #8 : Variance

Example Question #1 : Standard Deviation

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