Arithmetic mean:        given: 

Solve:  

The total score of those 9 students is .

The standard score for the entire class was .

Therefore, the last person needs a score of .

By adding all of the expressions we get  

which averages to ,

which will simplify to .

If the mean of 6 numbers is 15, that implies that the sum of the numbers is equal to 90 (6 x 15).

Therefore, or

Summing total money and dividing by the number of suspects gives us an average of approximately $2.73.  By comparing it to the amounts held by the suspects, we can see that Suspect #2 is guilty.

In order to calculate mean, we can act as though the temperature was exactly for the first six days and on the seventh day. The mean can be expressed:

In order to calculate the new average, we can act as though each of the thirteen students who originally took the test each earned 87%:

We are asked to find out how much Tom would win on average.  Since Bob has a 75% chance to win, Tom must have a 25% chance to win.  The amount Tom will lose is given to us: $100.  We now need to find how much he will win.

The first day, Bob will only pay $1.  Similarly for the second day. The third day Bob will now have to pay $2.  The fourth day, $4.  We notice that each day we will be doubling how much he pays (since he is matching how much was paid previously).  Thus he is paying:

Now, there are a couple ways to deal with this series.  There are only 10 terms, so we can simply add them up to get our answer. Another method would be looking at it as a geometric series.  If you recall, the sum of a geometric series (series that have terms differentiated by a constant multiplier) is given by:  where  is the initial value,  is the ratio between any 2 terms, and is the number of terms.  Here we just have , since the first term is a 1.  Each term is half the previous term, so , and we have 9 terms ( to ) so .

Hence this would be: 

and our answer would be 

We now have that Bob will have to pay $512 if Tom wins, while Tom will have to pay $100 if Bob wins.  We finally take into account the odds of winning.  25% of the time, Tom will win $512.  75% of the time he will lose $100.  We can more easily interpret this with fractions: 1/4 chance to win, 3/4 chance to lose.

Now we simply keep in mind how averages work: The numerator is going to be the sum of all the cases, while the denominator is the number of cases.  In this situation, we have 4 cases.  3 of them Tom loses, 1 he wins.  Thus we have:

So on average, Tom will make $53 with this bet.  The closest answer to this is $50.

List the numbers in order to find the median:  {50, 54, 54, 58, 59, 61, 63, 66, 67, 68}.  The middle numbers are 59 and 61 so their average is 60 for the median of the data set making statement I true.  Add the list of numbers and divide by 10 to get a mean of 60, making statement II true.  There is a mode of 54 so statement III is false.