Data Analysis and Probability - ISEE Upper Level Mathematics Achievement

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Question

Consider the following set of scores from a math test. What is the mean of these scores?

$\small \small \left { 70,89,67,77,92,83,67,75\right }$

Answer

To find the mean, first sum up all the values.

$\small \sum \left { 70,89,67,77,92,83,67,75\right }=620$

The divide the result by the number of values

$\small 620\div8=77.5$

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Question

Alice's numeric grade in her physics class is determined by four equally weighted hourly tests and a final, weighted twice as much as an hourly test. The highest score possible on each is 100.

Going into finals week, Alice's hourly test scores are 76, 85, 82, and 87. What must Alice make on her final, at minimum, in order to average 90 or more for the term?

Answer

Alice's grade is a weighted mean in which her hourly tests have weight 1 and her final has weight 2. If we call her final, then her term score will be

$\frac{76 + 85 + 82 + 87 + 2x}{1 + 1 + 1 + 1 + 2}$ ,

which simplifies to

$\frac{2x+330 }{6}$ .

Since she wants her score to be 90 or better, we solve this inequality:

$\frac{2x+330 }{6}\geq 90$

$\frac{2x+330 }{6} \cdot 6\geq 90\cdot 6$

$2x+330\geq540$

$2x+330 -330\geq540-330$

$2x\geq 210$

$2x\div 2 \geq 210 \div 2$

$x\geq 105$

Since it is established that 100 is the maximum score, Alice cannot achieve an average of 90 or more for the term.

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Question

Sally's numeric grade in her economics class is determined by four equally weighted hourly tests, a midterm weighted twice as much as an hourly test, and a final weighted three times as much as an hourly test. The highest score possible on each is 100.

Going into finals week, Sally's hourly test scores are 89, 85, 84, and 87, and her midterm score is 93. What must Sally make on her final at minimum in order to average 90 or more for the term?

Answer

Sally's grade is a weighted mean in which her hourly tests have weight 1, her midterm has weight 2, and her final has weight 3. If we call her score on the final, then her course score will be

$\frac{89 + 85 + 84 + 87 + 2 \cdot 93 + 3x}{1 + 1 + 1 + 1 + 2 + 3}$ ,

which simplifies to

$\frac{89 + 85 + 84 + 87 + 186 + 3x}{9} = \frac{3x + 531}{9}$ .

Since Sally wants at least a 90 average for the term, we can set up and solve the inequality:

$\frac{3x + 531}{9} \geq 90$

$\frac{3x + 531}{9} \cdot 9 \geq 90 \cdot 9$

$3x + 531 \geq 810$

$3x + 531-531 \geq 810-531$

$3x \geq 279$

$3x \div 3 \geq 279 \div 3$

$x \geq 93$

Sally must score at least 93 on the final.

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Question

Fred's course average in French class is the average of the best five of his six hourly test scores. Going into finals week, Fred has scores of 78, 77, 84, 89, and 72. How much, at minimum, must Fred score on his sixth test in order to make an average of 80 or better for the term?

Answer

If Fred does not take the sixth test or gets a 0 on it, he will receive the average of his first five tests. This is

$\frac{78 + 77 + 84+ 89 + 72}{5} =\frac{400}{5} = 80$ .

Since he can only improve his class grade by taking the sixth test, Fred is already assured of an average of 80 or better.

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Question

The mean of six numbers is 77. What is their sum?

Answer

The mean of six numbers is their sum divided by 6, so the sum is the mean multiplied by 6. This is:

$77 \times 6 = 462$

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Question

Eric has taken four tests in his English class. His scores on the tests are 94, 87, 95, and 89. The upcoming final exam will be weighted as much as each of the previous four tests. What is the lowest score Eric can get on the final exam and still have a mean score of no less than 90?

Answer

The first step here, since you're working with the average (or mean), is to determine what is the sum of five test scores that will result in Eric earning at least a mean score of 90.

Start with the fact that he will need to make at least a 90 for his average score. Since you already have the average score, you're working backwards. You need to multiply 90 by 5 for the 5 test scores that will be used to get that average:

$90\times 5=450$

This tells you that the five scores - the four test scores and the final exam - will need to add up to at least 450 in total.

You know the four original test scores, so you can add those up:

So you know the four test scores add up to 365. To get an average of at least 90, the fifth score must bring that sum to at least 450. To find the minimum final exam test score, you subtract 365 from 450:

This tells you that Eric must score at least an 85 on the final exam to achieve an overall mean score of 90.

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Question

Consider the time Becky spends doing homework each night. What is the average number of hours Becky spends on homework each night (consider only Monday-Thursday)?

Monday: 3.2 hours

Tuesday: 1.1 hours

Wednesday: 2.7 hours

Thursday: 0.9 hours

Answer

To calculate the average, find the sum of the total hours divided by the number of nights:

$avg=\frac{3.2+1.1+2.7+0.9}{4}=\frac{7.9}{4}=1.975$

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Question

The mean of the following numbers is . Solve for .

Answer

The mean is determined by dividing the sum of the values by the total number of values:

$mean=\frac{17+14+22+11+14+16+n}{7}=15$

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Question

What is the mean of the first ten prime numbers?

Answer

The first ten primes form the data set:

$\left { 2, 3, 5, 7, 11, 13, 17, 19, 23, 29\right }$

Add them, and divide by :

$\left ( 2 + 3+5+ 7+ 11+13+17+19+ 23+29 \right )\div 10 = 129\div 10= 12.9$

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Question

David's course score in a chemistry course is the mean of five tests. He has scored and . What must he score on the fifth test to be assured of a course score of ?

Answer

Let be the minimum fifth test score. Then the average of the scores is

$\frac{N + 72 + 81 + 79 + 85 }{5} = 80$ .

$\frac{N + 317 }{5} = 80$

$\frac{N + 317 }{5} \cdot 5 = 80 \cdot 5$

, the minimum score David needs.

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Question

A gymnastics meet has seven judges. After each routine, each judge assigns a merit-based score from 0 to 10; a contestant's score for the routine is the mean of all the judges' scores except for the highest and the lowest.

The seven judges individually assigned the following scores to one of Donna's routines: . What is Donna's score, rounded to the nearest tenth?

Answer

The highest and lowest scores, and , are tossed out, so Donna's score is the mean of the five that remain.

$\frac{ 9.2+ 9.5+9.6+ 9.5+ 9.0 }{5} = \frac{ 46.8 }{5} = 9.36 \approx 9.4$

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Question

The heights of a football team's members are and inches. Find the mean of the heights.

Answer

The mean is the sum of all the data values divided by the number of values. So we can write:

$Mean=\frac{68+70+75+72+73+78}{6}\approx 72.667 in$

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Question

Consider the following set of scores from a physics test. Give the mean of the scores.

$\left { 75,73,84,81,68,95,77 \right }$

Answer

The mean of the scores can be calculated as:

$\bar{x}=\frac{1}{n}\sum_{i=1}^n{x_{i}$

Where:

$\bar{x}$ is the mean of a data set, $\sum_{}^{}$ indicates the sum of the data values $x_{i}$ , and is the number of data values. So we can write:

$\bar{x}=\frac{1}{n}\sum_{i=1}^n{x_{i}=\frac{75+73+84+81+68+95+77}{7}=\frac{553}{7}=79$

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Question

The mean of five numbers is . Give their sum.

Answer

The mean of five numbers is equal to their sum divided by , or as a formula we can write:

$\bar{x}=\frac{1}{n}\sum_{i=1}^n{x_{i}$

Where:

$\bar{x}$ is the mean of a data set, $\sum_{}^{}$ indicates the sum of the data values $x_{i}$ , and is the number of data values. So we can write:

$\sum_{i=1}^n{x_{i}=n\bar{x}\Rightarrow \sum_{i=1}^n{x_{i}=5\times 32=160$

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Question

The sum of numbers is . Find the mean in terms of . ( is a real number)

Answer

The mean of eight numbers is equal to their sum divided by or as a formula we can write:

$\bar{x}=\frac{1}{n}\sum_{i=1}^n{x_{i}\Rightarrow \frac{8t+16}{8}=\frac{8(t+2)}{8}=t+2$

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Question

In the following set, find the mean in terms of .

$\left { t^2-1,t,t^2,2t,1-t^2,-t^2,7t,8-2t \right }$

Answer

The mean of the scores can be calculated as:

$\bar{x}=\frac{1}{n}\sum_{i=1}^n{x_{i}$

Where:

$\bar{x}$ is the mean of a data set, $\sum_{}^{}$ indicates the sum of the data values $x_{i}$ , and is the number of data values. So we can write:

$\bar{x}=\frac{1}{n}\sum_{i=1}^n{x_{i}=\frac{t^2-1+t+t^2+2t+1-t^2-t^2+7t+8-2t}{8}$

$=\frac{(t^2+t^2-t^2-t^2)+(t^2+2t+7t-2t)+(-1+1+8)}{8}$

$=\frac{0+8t+8}{8}=\frac{8(t+1)}{8}$

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Question

The sum of a set of values is . If the mean of the data set is , give the number of values in the set.

Answer

Mean of the scores can be calculated as:

$\bar{x}=\frac{1}{n}\sum_{i=1}^n{x_{i}$

Where:

$\bar{x}$ is the mean of a data set, $\sum_{}^{}$ indicates the sum of the data values $x_{i}$ , and is the number of data values. So we can write:

$\bar{x}=\frac{1}{n}\sum_{i=1}^n{x_{i}\Rightarrow n=\frac{1}{\bar{x}}\sum_{i=1}^n{x_{i}=\frac{72}{12}=6$

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Question

Consider the following set of scores from a math test. If the mean is , give .

$\left { 80,75,74,70,90,84,t \right }$

Answer

Mean of the scores can be calculated as:

$\bar{x}=\frac{1}{n}\sum_{i=1}^n{x_{i}$

Where:

$\bar{x}$ is the mean of a data set, $\sum_{}^{}$ indicates the sum of the data values $x_{i}$ , and is the number of data values. So we can write:

$\bar{x}=\frac{1}{n}\sum_{i=1}^n{x_{i}$

$\Rightarrow 6=\frac{80+75+74+70+90+84+t}{7}$

$\Rightarrow 6=\frac{473+t}{7}\Rightarrow 473+t=6\times 7$

$\Rightarrow t=42-473=-431$

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Question

The mean of test scores is . When the student takes the test, the mean goes down by two points. Give the score.

Answer

The total of test scores is $14\times 75=1050$ . The mean of the total scores for students would be:

$\bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_{i}$

Where:

$\bar{x}$ is the mean of a data set, $\sum$ indicates the sum of the data values $x_{i}$ and is the number of data values. So we can write:

$\bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_{i}=\frac{1050+x_{15}}{15}=75-2$

$\Rightarrow 1050+x_{15}=73\times 15$

$\Rightarrow x_{15}=1095-1050$

$\Rightarrow x_{15}=45$

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Question

The sum of test scores is . When the student takes the test and gets a score of , how does the mean change?

Answer

The mean is the sum of the data values divided by the number of values or as a formula we have:

$\bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_{i}$

Where:

$\bar{x}$ is the mean of a data set, $\sum$ indicates the sum of the data values $x_{i}$ and is the number of data values. The total of test scores is ; the mean of the scores is:

$\bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_{i}=\frac{1558}{19}=82$

When the student takes the test the sum of scores would be and the new mean of the scores is:

$\bar{x_{1}}=\frac{1}{n}\sum_{i=1}^{n}x_{i}=\frac{1628}{20}=81.4$

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