! www.clutchprep.com

!

STATISTICS - CLUTCH

CH.6: TYPES OF CONTINUOUS RANDOM VARIABLE DISTRIBUTIONS

PROBABILITY DENSITY CURVES

Continuous variables are quantitative variables that cannot be broken down into ________________

Anything measured: weights, volume, money, time, etc.

As a discrete variable gets broken down into smaller and smaller chunks, the histogram begins to look _________

When a variable is continuous the smooth curve associated with it is called a ________________

The area under the density curve is the probability of an event

CONTINOUS DISTRIBUTIONS

Two key characteristics of the continuous distribution are:

(1) there’s a specific _________ of possibilities

(2) all intervals in the distribution are ___________ probable

Earlier we said that the probability is simply the area under the curve

Probability =

-a

EXAMPLE 1: iPhones are expected to have a attery life of etween 0 and 6 hours. Assuming that there’s an equal

chance that an iPhone’s battery life is somewhere between this interval, what is the probability that you purchase an iPhone

with 14 hours of battery life?

EXAMPLE 2: Referring back to Example 1, what is the probability that you purchase an iPhone with a battery life between

12 and 14 hours?

Pro

bab

ility

Pro

bab

ility

STATISTICS - CLUTCH

CH.6: TYPES OF CONTINUOUS RANDOM VARIABLE DISTRIBUTIONS

Page 2

PRACTICE 1: Suppose that a child’s weight is never lower than 6 pounds, nor higher than 10 pounds and each weight is

equally likely. What is the probability of giving birth to a child who weighs exactly 9.5 pounds?

PRACTICE 2: Referring to Practice 1, what is the probability of having a child who weighs between 6 and 9 pounds?

PRACTICE 3: Referring to Practice 1, what is the probability of having a child who weighs between 9 and 12 pounds?

PRACTICE 4: Assuming that it is equally likely that you end up with a 10-13” foot-long sub from Subway, what is the

probability that you get a foot-long sub that is only 10” long?

PRACTICE 5: Referring to Practice 4, what is the probability that you get a foot-long that is 10-13” long?

STATISTICS - CLUTCH

CH.6: TYPES OF CONTINUOUS RANDOM VARIABLE DISTRIBUTIONS

Page 3

CHEBYSHEV’S RULE

For any distribution, regardless of the shape, chebyshev’s ________ helps you make estimations for probabilities using

the ____ as a ruler

Between µx + σx and µx – σx = ___________ observations

Between µx + 2σx and µx – 2σx = at least _________ of observations

Between µx + 3σx and µx – 3σx = at least _________ of observations

In general, the proportion of observations within any interval between µx + kσx and µx – kσx :

EXAMPLE 1: For the following data set, what is the interval for which at least ¾ of the observations lie?

1 3 5 7 9 11

EXAMPLE 2: Given that a data set has a mean and standard deviation of 10 and 2.5, respectively, how many observations

are expected to lie within 3.75 and 16.25?

P(x) = at least 1 – 1

k2

x = observation

μx = mean

σx = SD

k = constant

STATISTICS - CLUTCH

CH.6: TYPES OF CONTINUOUS RANDOM VARIABLE DISTRIBUTIONS

Page 4

PRACTICE 1: For the following data set, at least what percent of observations would you expect to lie within 6.16 and -.16?

1 2 3 4 5

PRACTICE 2: Referring to Practice 1, between what two values would you expect 8/9 of the observations to lie?

PRACTICE 3: Referring to Practice 1, between what two values would you expect 84% of observations to lie?

PRACTICE 4: Assuming that a particular data set has a mean and variance of 100 and 144, respectively, at least what

percent of observations would you expect to see between 88 and 112?

PRACTICE 5: Referring to Practice 4, what is the probability of finding an observation that lies between 82 and 118?

STATISTICS - CLUTCH

CH.6: TYPES OF CONTINUOUS RANDOM VARIABLE DISTRIBUTIONS

Page 5

NORMAL DISTRIBUTIONS

The normal distribution is another form of a _________________ distribution

A normal distribution has three key qualities:

(1) bell shaped

(2) symmetric to the ______________

(3) the area under the curve is _________________

EMPIRICAL RULE

The empirical helps you make estimations for probabilities using the ____ as a ruler

Between µx + σx and µx – σx = _________ of observations

Between µx + 2σx and µx – 2σx = _________ of observations

Between µx + 3σx and µx – 3σx = _________ of observations

EXAMPLE 1: There’s a myth that guy’s shoe sizes are proportional to other body parts…Assuming the mean and standard

deviation of men’s shoe sizes are 9 and 2 respectively, what proportion of men should have shoe sizes between 7 and 11?

EXAMPLE 2: Women’s shoe sizes average a 7 with a standard deviation of 1. What is the probability of randomly selecting

a woman with a shoe size between 8 and 9?

μx = mean

σx = SD

STATISTICS - CLUTCH

CH.6: TYPES OF CONTINUOUS RANDOM VARIABLE DISTRIBUTIONS

Page 6

PRACTICE 1: The standard shot is supposed to be 1.5 oz. The average shot actually ends up being 1.7 oz. with a standard

deviation of .1 oz. What proportion of shots will fall between 1.5 and 1.9 oz.?

PRACTICE 2: Referring to Practice 1, what is the probability that a shot will be between 1.9 and 2.0 oz?

PRACTICE 3: The standard double shot is supposed to be 2.5 oz. The actual shot, on average, is 2.8 oz. with a standard

deviation of .3 oz. What percentage of double shots lies between 2.5 and 3.1 oz.?

PRACTICE 4: Referring to Practice 3, what is the probability that a double shot will be between 2.2 and 2.8 oz.?

PRACTICE 5: Draw the normal distributions referring to Practice 1 and Practice 3 along a number line.

STATISTICS - CLUTCH

CH.6: TYPES OF CONTINUOUS RANDOM VARIABLE DISTRIBUTIONS

Page 7