Statistics

The 68-95-99.7 rule is applicable for a Normal distribution.

The Normal Distribution is determined by two parameters: the mean (m) and the standard deviation (s). They are normally denoted by Greek letters but there is no telling what this rubbish browser will do to them.

Anyway, these are used to calculate z-scores for observations. The z score for an observation x is (x - m)/s.

The rule is that

68% of the observation lie within -1 < Z < 1 : that is within 1 standard deviation of the mean;

95% of the observation lie within -2 < Z < 2 : that is within 1 standard deviation of the mean;

68% of the observation lie within -3 < Z < 3 : that is within 1 standard deviation of the mean.

Also, the Normal distribution is symmetrical so 50% of the observations are for z < 0.

Now, 1 - 68% is outside of -1 < Z < 1. That is, the two regions Z < -1 and Z > 1 = 32% and so, by symmetry, each is half as much which is 16%. Thus the first of the above rule also implies the following probabilities:

• 16% for Z < -1 and so 84% for Z > -1
• 16% for Z > 1 and so 84% for Z < 1

Now, given that m = 70 and s = 15

(a) X < 55 => Z < (55 - 70)/15 = -1

So prob (X < 55) = 0.16

(b) X < 40 => Z < (40 - 70)/15 = -2

and so, prob(X < 40) = 2.5%

(c) X < 85 => Z < 1

S0 prob (X < 85) = 84%

(d) Prob(X < 100) = 97.5%

(e) Prob(X > 85) = 16%

Each roll has (6 x 6) = 36 possible outcomes.

A 7 or 11 can be rolled in 8 ways:

1--6, 6--1

2--5, 5--2

3--4, 4--3

5--6, 6--5

The probability is (8/36) = 2/9

Doubles can be rolled in 12 ways:

1--1 and reverse

2--2 and reverse

3--3 and reverse

4--4 and reverse

5--5 and reverse

6--6 and reverse

The probability is (12/36) = 1/3.

The probability of both of these rolls happening consecutively is (2/9 x 1/3) = 2/27 = 7.41% (rounded)

There are three types of correlation: positive, negative, and none (no correlation).

• Positive Correlation: as one variable increases so does the other. Height and shoe size are an example; as one's height increases so does the shoe size.
• Negative Correlation: as one variable increases, the other decreases. Time spent studying and time spent on video games are negatively correlated; as the your time studying increases, time spent on video games decreases.
• No Correlation: there is no apparent relationship between the variables. Video game scores and shoe size appear to have no correlation; as one increases, the other has no effect. A No Correlation graph would show this.

Each roll of the coin can either have a head or not have a head. So in three rolls there could be:

• no heads (all tails);
• 1 head (either first, second or third roll was the head, the other two being tails);
• 2 heads (either first, second or third roll was the tail, the other two being heads); or
• 3 heads (all heads)

giving a sample space for the number of heads in three rolls of a coin:

{0, 1, 2, 3}