Algebra II Regents Tutoring Topics

SECTION: Statistics

__EXAMPLE 1:__

To answer this question you need to know what is the “variability of data” in this context?

Standard deviation, sigma tells us how deviated the data is from the mean. The higher the standard deviation is the more spread out the data is. How come this is true? Imagine a bar graph with each bar representing one standard deviation. In the middle of the bar graph is the mean, or the average. The higher the number of each bar the wider each bar is (farther away from the center) and more wider bars will result in a spread out graph.

What does it mean to have more variability?

**Variability** is the extent to which data points in a statistical distribution or data set diverge from the average, or **mean**, value as well as the extent to which these data points differ from each other. There are four commonly used measures of **variability**: range, **mean**, **variance** and standard deviation.

If the standard deviation is higher for the second data set that means the variability is greater for that data set.

Choice (2)

__EXAMPLE 2:__

What are independent events?

Independent events are * independent *of each other. One event has not bearing on the other. The probabilities of those events have no effect on each other. The outcome of one event is not effected by the outcome of the other event.

** **

**-Mutually exclusive events are not the same as independent events. **

**-Disjoint events are the same as mutually exclusive. **

** **

mutually exclusive events(disjoint events) – events that can not happen at the same time.

disjoint events – if one event occurs the other one can not.

independent events – each event has not effect on the other’s probability.

**-Event intersection is zero for mutually exclusive events. **

**(Because those events cannot occur together**

**P(A and B) =0)**

If two events are disjoint they are dependent, because if one event occurs the other one can not has an impact on the other event. Therefore the fact that one event cannot occur because of the other means that one event has an effect on the other and therefore is dependent on the other not occuring.

What is the formula for P(A and B)?

Eliminate Choices (1) (3) (4).

The multiplication of P(A)*P(B) does not yield 0.4

Choice (2).

How many students will be enrolled in a tutoring program?

The number of students who score less then one standard deviation above the mean will be enrolled.

__REMINDER:__

One standard deviation above the mean means one bar(interval) to the right from the mean. Each bar represents one standard deviation in a normally distributed distribution (check out the graph below !>).

68 – MEAN

One S.D. = 7.2 – the equivalent of ONE INTERVAL

How many students are below one standard deviation?

Well based on the graph 50 % of the students are below the mean, and 50 % of the students are above the mean.

Which means 50% + 34.1% = 84.1% of the students are below one standard deviation.

How much is 84.1% out of all the students?

750 * 0.841 = 630.75.

Choice (1)

Basic conditional probability question.

__NOTE:__

If the question contains the word probability “given” some other probability. It is a conditional probability question.

__GIVENS:__

__ __

P(O) = 0.8

P(F | O)= 0.85

P( O and F) =?

P (F | O ) = P(F and O) / P(O)

0.85 = P (F and O) / 0.8

multiply both sides by 0.8

0.68 = P ( F and O )

__QUESTION: __

Is P (F and O ) = P (O and F).

Another words is the intersection of F and O the same as the intersection of O and F. Yes it is.

__QUESTION:__

In your conditional probability formula can A and B be switched? What do I need to remember when remembering to use the formula?

__THINGS TO REMEMBER:__

P(A | B) – -> always reads P of A given B this can not be switched.

P(A | B) * P (B) = P (the intersection of A and B)

The conditional Probability is multiplied by the second Probability to give P (intersection). This order cannot be switched.

__ __

The simulation graph shows the average (mean = 29.101).

Interval containing middle 95 % of the data is two standard deviations below the mean + two standard deviations above the mean.

Based on the simulation picture the majority of the data (95 %) falls between 27.0 and 31.0 __on the graph. Let’s calculate it using the values provided rounding to the nearest hundredth.__

__ __

If you add two standard deviations to the mean 29.101 + 2(0.934) = 27.23

If you subtract two standard deviations from the mean you get the left bound.

29.101 – 2(0.934) = 27.23

**(27.23, 30.97)—The 95 % confidence interval, means that there is a 95% chance that the mean will fall within this range.**

**Therefore it is possible that the mean is 30.**

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