1) Each of the three measures of central tendency?the mean, the median, and the mode?are more appropriate for certain populations than others.

For each type of measure, give two additional examples of populations where it would be the most appropriate indication of central tendency.

The mean (average) of a population is most appropriate when the data are normal or symmetrically distributed, and there are no outliers. When there are outliers, the mean might be inappropriately high or low. You could use the mean when measuring things like height or weight, as long as there aren't any outliers.

The median (middle value) of a population is most appropriate when the data is not symmetrical or there are outliers. Outliers don't affect the value of the median as much as they affect the value of the mean. Often, you use the median for things like income or temperature, which tend to have outlying values.

The mode (the number that occurs the most frequently) is not often used in statistics. It would be appropriate in a situation such as asking consumers which product they liked the best - you wouldn't be able to average the results (out of products 1, 2, and 3, there is no product 1.5, but it would be useful to say that people chose product 3 more times than they chose the others). It would also be useful if one number showed up many more times than the others in a data set.

2. Find the mean, median, and mode of the following data set:
5 15 9 22 67 42 2 72 81 53 6 70 41 9 42 23

First put the data in order:
2 5 6 9 9 15 22 23 41 42 42 53 67 70 72 81

The average is 34.9375:

(2 + 5 + 6 + 9 + 9 + 15 + 22 + 23 + 41 + 42 + 42 + 53 + 67 + 70 + 72 + 81)/16 = 559/16 = 34.9375

The median is 32 (take the average of the two middle numbers):

2 5 6 9 9 15 22 23 41 42 42 53 67 70 72 81

(23 + 41)/2 = 32

The modes are 9 and 42 (they both show up twice):

2 5 6 9 9 15 22 23 41 42 42 53 67 70 72 81

3. Sometimes, we can take a weighted approach to calculating the mean. Take our example of high temperatures in July.

Suppose it was 98°F on 7 days, 96°F on 14 days, 88°F on 1 day, 100°F on 6 days and 102°F on 3 days.

Rather than adding up 31 numbers, we can find the mean by doing the following:
Mean = ( 1 x 88 + 14 x 96 + 7 x 98 + 6 x 100 + 3 x 102) / 31
...where 1, 14, 7, 6, and 3 are the weights or frequency of a particular temperature's occurrence.

Then we divide by the total of number of occurrences.

Suppose we are tracking the number of home runs hit by the Boston Red Sox during the month of August:

Number of Games HRs Hit each Day
2 3
5 2
6 1
7 0

Using the weighted approach, calculate the average number of home runs per game hit by the Sox.

(2*3 + 5*2 + 6*2 + 7*0)/(2 + 5 + 6 + 7)
= (6 + 10 + 12)/20
= 28/20
= 1.4 home runs per game

4. When a pair of dice is rolled, the total will range from 2 (1,1) to 12 (6,6). It is a fact that some numbers will occur more frequently than others as the dice are rolled over and over.

A. Why will some numbers come up more frequently than others?

Some numbers can be made in more ways than others. You can only roll a two in one way (1 and 1), but you can roll a six in multiple ways (1 and 5, 2 and 4, 3 and 3, 4 and 2, 5 and 1).

B. Each die has six sides numbered from 1 to 6. How many possible ways can a number be rolled? In other words, we can roll (2,3) or (3,2) or (6,1) and so on. What are the total (x,y) outcomes that can occur?

The first die can have 6 possibilities; same for the second die. Therefore there are (6)(6) = 36 total outcomes.

C. How might you then estimate the percentage of the time a particular number will come up if the dice are rolled over and over?

You would look at how many ways a number could be made rolling the dice, then divide that number by 36. For example, in part A we saw that a 2 can only be rolled in one way, so the probability of rolling a 2 is 1/36 = 0.0278. Same for 12, because it can only be made by rolling two 6's. But, a 6 can be rolled in five ways, so the probability of rolling a 6 is 5/36 = 0.1389.

D. Once these percentages have been calculated, how might the mean value of the all the numbers thrown be determined?

You use a version of the weighted average approach: you add together the values multiplied by their respective probabilities.

(2)(0.0278) + ... + (6)(0.1389) + ... + (12)(0.0278)

E. If you have completed the Discussion Board assignment, you have an idea of what a population distribution is. There is a very famous distribution that describes the frequency of the number of times a number comes up in a series of dice rolls. Use the Library or the Internet to see if you can find its name.

It is the Binomial Distribution (also called the Bernoulli Distribution). This distribution describes the probabilities of a "success" (for example, we could call rolling a 6 a success) happening different numbers of times if you know the probability of a success happening once. You would use it to answer the question: if you roll a pair of dice 100 times, what is the probability of rolling a 6 only once? Twice? 50 times? etc.

please see the attached file.