Statistics and Probability - Algebra

Card 0 of 6336

Question

Actuaries (people who determine insurance premiums for things like life and car insurance) often have to look at the average insurance costs in an area. One way to do this without letting outliers affect their data is to take the standard deviation of insurance costs in an area over a given period of time.

Calculate the standard deviation from the data set of insurance claims for a region over one-year periods (units in millions of dollars). Round your final answer to the nearest million dollars.

Answer

The first step in calculating standard deviation, or $\sigma$ , is to calculate the mean for your sample, or $\mu$ . Remember, to calculate mean, sum your data values and divide by the count, or number of values you have.

$\mu= \frac{x_{1} +x_{2} + ...x_{n}}{n} = \frac{32+38+16+21+25+28+23+31+16+25+32}{11} \approx 26.1$

Next, we must find the difference between each recorded value and the mean. At the same time, we will square these differences, so it does not matter whether you subtract the mean from the value or vice versa.

We use $\sum_{i=1}^{N} (x_{i}-\mu)^{2}$ to represent this, but all it really means is that you square the difference between each value $x_{i}$ , where is the position of the value you're working with, and the mean, $\mu$ . Then we sum all those differences up (the part that goes $\sum_{i=1}^{N}$ , where is your count. $_{i=1}$ just refers to the fact that you start at the first value, so you include them all.)

It's probably easier to do than to think about at first, so let's dive in!

$(32-26.1)^{2} = 34.81$

$(38-26.1)^{2} = 141.61$

$(16-26.1)^{2} = 102.01$

$(21-26.1)^{2} = 26.01$

$(28-26.1)^{2} = 3.61$

$(23-26.1)^{2} = 9.61$

$(31-26.1)^{2} = 24.01$

$(16-26.1)^{2} = 102.01$

$(32-26.1)^{2} = 34.81$

Now, add the deviations, and we're nearly there!

$\sum_{i=1}^{N} (x_{i}-\mu)^{2} = 34.81 + 141.61 + 102.01 + ... + 34.81 = 480.9$

Next, we must divide this number by our :

$\frac{480.9}{11} = 43.72$

This number, 43.35, is our variance, or $\sigma^{2}$ . Since standard variation is $\sigma$ , you may have guessed what we must do next. We must take the square root of the summed squares of deviations.

$\sigma = \sqrt{\frac{1}{N}\sum_{i=1}^{N} (x_{i}-\mu)^{2}} = \sqrt{43.72} = 6.612$

So, our standard deviation is 7 million dollars (remembering to round to the nearest million, per our instructions.)

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Question

Find the mean of the following set of numbers:

Answer

To solve, you must sum up the numbers and divide by the quantity. Thus,

$mean=\frac{1+3+4+7+10+13+16+17+19}{9}=10$

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Question

Find the mean of the following numbers (round to the nearest tenth).

Answer

Add every number together and divide by the total number of numbers in the set (9) to get the following.

$\ \frac{24+ 35+ 29+ 49+ 39+ 29+ 24+ 24+ 30}{9}\ \=\frac{283}{9}\ \=31.4$

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Question

Find the median of this set of numbers.

Answer

Rearrange the numbers into increasing order.

$134, 143, 235, 243, {\color{Red} 243}, 346, 724, 745, 823$

The number in the center of the set is the median: 243

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Question

Find the mode of the following set of numbers: .

Answer

The mode of any set of numbers is the number in the set that appears most often. In order to find the mode, it is easier to put the numbers in the set in order from least to greatest. The set we were given is

.

In order from least to greatest, the set is:

Next, we will look through our sequenced set to see if any numbers appear more than one time. In this case, there are several numbers that appear more than once: , , and all appear more than once in our set.

Since we have multiple numbers that appear more than once, now we will look to see how many times each of these numbers appears in the set.

- appears two times

- appears two times

- appears four times

Since appears more times in the set than any other number, is the mode of our set.

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Question

Refer to the following set of numbers:

Find the mode of the set.

Answer

The mode of the set is the most repeated number. In this case is the mode because it is in the set 3 times.

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Question

Refer to the following set of numbers:

Find the mode of the set.

Answer

The mode of the set is the most repeated number. In this case is the mode because it is in the set 3 times.

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Question

Refer to the following set of numbers:

Find the mode of the set.

Answer

The mode of the set is the most repeated number. In this case is the mode because it is in the set twice.

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Question

What is the mode of the following set of numbers?

Answer

We define mode as the number that appears most often in a set of numbers. Examining the set above, we see that no number appears more times than any other number, and so the set has no mode.

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Question

Find the mode of the following set of numbers:

Answer

Find the mode of the following set of numbers:

Mode is simply the most repeated number in the set.

In this case, we have three 34's, so our answer must be 34. Nothing else comes close.

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Question

Find the mode of the following data set:

3, 9, 5, 4, 4, 10, 9, 2, 3, 1, 0, 9

Answer

The mode of a data set is simply the number that occurs most frequently. In this case, the number 9 occurs most frequently, appearing 3 times.

Therefore, the mode of the given data set is 9.

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Question

Find the mode of the following data set:

13, 4, 6, 3, 9, 3, 11, 11, 3

Answer

The mode of a data set is the number within the set that appears most frequently Given the data set

13, 4, 6, 3, 9, 3, 11, 11, 3

The number 3 appears most frequently. Therefore, the mode is 3.

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Question

Find the mode of the following set:

Answer

To find the mode of any set, find the number which occurs the greatest number of times in the set. The best answer is eight.

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Question

What is the mean given the following set? $x=[\frac{1}{2}, \frac{1}{4}, \frac{1}{6}]$

Answer

To find the mean, add all the numbers in the set and divide by the total numbers in the set.

Convert $\frac{1}{2}+\frac{1}{4}+\frac{1}{6}$ to a common denominator and then add to find the total.

$\frac{1}{2}+\frac{1}{4}+\frac{1}{6} = \frac{6}{12}+\frac{3}{12}+\frac{2}{12} = \frac{11}{12}$

Dividing this sum by three is the same as multiplying the sum by one third.

$\frac{11}{12} \div 3 = \frac{11}{12}\times \frac{1}{3} = \frac{11}{36}$

The average is $\frac{11}{36}$ .

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Question

Find the mode of the following set:

Answer

To find the mode of any set, find the number which occurs the greatest number of times in the set. The best answer is one.

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Question

Refer to the following set of numbers:

Find the median of the set.

Answer

The median of the set is the middle number. To find this, we order the numbers from smallest to greatest.

In this case, the middle number is

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Question

Refer to the following set of numbers:

Find the range of the set.

Answer

To find the range of the set, subtract the smallest number in the set from the largest one.

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Question

Refer to the following set of numbers:

Find the median of the set.

Answer

The median of the set is the middle number. To find this, we order the numbers from smallest to greatest.

In this case, the middle number is

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Question

Find the median of the following numbers:

{1, 9, 3, 14, 15, 13, 2, 7, 8, 4, 5}

Answer

Start by putting the numbers in ascending order:

{1, 2, 3, 4, 5, 7, 8, 9, 13, 14, 15}

Once you do that you make your way to the middle number (as that is what the median is) which you find to be .

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Question

Find the mean of the following set of numbers (round to the nearest tenth):

Answer

To calculate the mean of any set of n numbers, add each of the numbers together, then divide the sum by n.

The best answer is:

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