Chpt 6 - Continuous Random Variables Flashcards
What is a continuous random variable?
A random variable that cannot be listed, the options are infinite
Generally, I would call them not whole numbers…so how tall are you?
I’m not 5 feet, I’m 5’8” or my weight, even for my personal records isn’t to the whole pound, we use half pounds on nearly all scales
What is a density curve?
A curve showing the distribution of the continuous random variable.
It’s basically created by connecting the tops of the bars of a histogram
What characteristics of a density curve lend itself to a bell-shaped curve?
Unimodal and symmetric
If the distributions of a random continuous variable is unimodal and symmetric, how can we describe the density curve?
Bell-shaped
What shape characteristics are we most concerned about when discussing the shapes of histograms or density curves?
Symmetry, skewness, and modality
What are the 2 parameters that determine a normal distribution?
Mean - a normal curve of a random variable is symmetric about and centered at it’s mean
Standard Deviation - determines the spread of a normal density curve
When looking at a density curve with a μ of 2, where would we expect the peak to be if it is a normal random variable?
2
If a continuous random variable has a bell-shaped density curve, what is the distribution of this random variable called?
Normal distribution
If a continuous random variable has a bell-shaped density curve, what is the random variable called?
Normal random variable
If a continuous random variable has a __________ density curve, the distribution of this random variable is called normal distribution and this random variable is called a normal random variable.
Bell-shaped
How would a normal distribution curve with a σ of 2 compared to another with a σ of 1?
The σ =2 would have
- smaller spread
-values concentrated around μ
-higher peak
How would a normal distribution curve with a σ of 1/2 compared to another with a σ of 1?
The σ = 1/2 would have
- wider spread
-values not concentrated around μ
-flatter peak
What is the notation for a normal random variable?
N(μ, σ)
What is the notation to describe a random variable that has a normal distribution?
X ~ N(μ, σ)
What are the characteristics of the density curve of a standard normal random variable?
μ = 0
σ - 1
A normal random variable with a mean of 0 and standard deviation 1 is called _________. It is denoted as ______. It’s distribution is called a ______.
Standard normal random variable.
Z
Standard normal distribution
If a normal random variable has a mean of 0 and a standard deviation of 1, what is it called?
Standard normal random variable
How is a standard normal random variable denoted?
Z
What is the distribution of a standard normal random variable called?
Standard normal distribution
What is the notation for a standard normal random variable?
Z ~ N (0, 1)
How do we find the area under a normal density curve?
First we transform it to a standard normal density and then we use the table II to find the area
How would I find the area to the left of the z-score using table II?
We look up the decimal values of the desired z-score and the number there is the area to the left of that z-score
How would I find the area to the right of the z-score using table II?
We look up the decimal values of the desired z-score and the number. This is the area to the left of that z-score, so we subtract this number from 1 so we get the area to the right of the z-score.
OR
We look up the decimal value on the opposite side of 0 of the desired z-score. So if I wanted the right side of 1.53, I’d look up the value for -1.53 and the number there is the area to the right. (Because it is symmetric to 0)
How do I look up the following value on table II?
P(Z<-2.52)
Look up the value of -2.52. Bam. Done.
How do I look up the following value on table II?
P(Z>-2.52)
Look up the value of +2..52. Bam. Done.
How do I look up the following value on table II?
P(Z<2.52)
Look up the value of +2..52. Bam. Done.
How do I look up the following value on table II?
P(Z>2.52)
Look up the value of -2.52. Bam. Done.
What is the total area under any density curve?
1
Say we want to find the probability that the value is between two values (a and b), what is the equation to solve this using table II?
P(a<Z<b) = P(Z<b) - P(Z<a)
When looking at a standard normal variable, what is the probability that the value of Z is between -1.75 (P=0.0401 and 2.72 (P=0.9967).
0.9967-0.0401= 0.9566
What is standardizing?
A mathematical operation that transforms a normal density curve and shifts it to a mean of 0 and standard deviation of 1
If you shift a density curve so that its center becomes 0 and standard deviation becomes 1 via mathematical operation, what is it called?
Standardizing
What is a normal random variable called after standardizing?
Standardized random variable
What is the equation for standardizing a normal random variable?
Z = X - μ
——-
σ
If P(x>a), what is the standardized z-score of a?
P(Z > a - μ
——-
σ )
If P(x<b), what is the standardized z-score of b?
P(Z < b - μ
——-
σ )
When evaluating the probability that a randomly selected female student is at most 158 cm high, what are we looking for on a density curve?
The left area of 158 under the density curve of height
OR
The area under the density curve of the random variable height and above the interval from the possible minimum height to 158 cm.
When evaluating the probability that a randomly selected female student is at least 167 cm high, what are we looking for on a density curve?
The right area of 167 under the density curve of height
OR
The area under the density curve of the random variable height and above the interval of 167 to the possible maximum height
When evaluating the probability that a randomly selected female student is between 150 and 166 cm high, what are we looking for on a density curve?
The area under the density curve of the random variable “height” and above the interval from the 150 cm to 166 cm.
First we find the values to the left of 150 and 166 and the difference between the two is the area between them.
What is a percentile? How many percentiles are there?
The percentiles of a random variable divide all the ordered observations of this random variable into hundredths, or 100 equal parts.
There are 99 percentiles
If we know the z-score, what is the equation to determine the x value?
x = z-score x σ + μ
What is the left area of the 10th percentile under the density curve of X?
How do we use this to find the z-score of this value?
0.1
Because there are 10% of observations of the random variable are smaller than the 10th percentile
We would look up this value using the middle values on table II. Once we find 0.1 (or the closest possible), we would look at the z-score who’s values are to the side and the top of the row to give us the z-score
If the bottom 1% of students will fail and exam, the minimum marks that a student has to score in order to pass is what percentile?
first
If the top 10% of students will get A+ on an exam, the minimum marks that a student has to score in order to get an A+ is what percentile?
90th percentile
This would leave 90% below and 10% above
A normal random variable X has a mean of 2 and a standard deviation of 0.5. The 10th percentile of this random variable has a z-score of -1.28. What is the equation to determine the 10th percentile of this random variable?
X = Z x σ + μ
= -1.28 x 0.5 + 2
If you have a calculator handy
X = 1.35
How would be find the z-score of the 20th percentile?
Look in the middle area for the value closest to 0.200
Look to the left for the first decimal place and the top row for the second decimal place
A normal random variable X has a mean of 5 and a standard deviation of 1. The 20th percentile of this random variable has a z-score of -0.84. What is the equation to determine the 20th percentile of this random variable?
X = Z x σ + μ
= -0.84 x 1 + 5
If you have a calculator handy
X = 4.16
Suppose that the mark of a randomly selected student in STAT 151 have a normal distribution with a mean of 70 and a standard deviation of 10.
The minimum mark a student needs to pass on an exam is the 1st percentile so the left area of its z-score under the standard normal density curve is 0.01. According to table II its z-score is -2.33.
What is the equation to determine the minimum mark a student needs to pass the exam?
X = Z x σ + μ
= -2.33 x 10 + 70
If you have a calculator handy
X = 46.7%
Suppose that the mark of a randomly selected student in STAT 151 have a normal distribution with a mean of 70 and a standard deviation of 10.
The minimum mark a student needs to get an A+ on an exam is the 90th percentile so the left area of its z-score under the standard normal density curve is 0.9. According to table II its z-score is 1.28.
What is the equation to determine the minimum mark a student needs to pass the exam?
X = Z x σ + μ
= 1.28 x 10 + 70
If you have a calculator handy
X = 82.8%
What is the notation for a z-score that has an area to the right under the standard normal curve?
How do we determine the z-score of this value?
z with a subscript α (alpha)
1-α = left area of z-score
Look up this number in the middle of table II, looking to the edges for the z-score
zα=0.025
What is the z-score for this area value?
1-0.025=0.975 if the left area of the z-score
We would look for 0.975 in the middle values of table II
It is 1.96
