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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?
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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