What is a continuous random variable ?

A continuous random variable is a random variable that can take any value within a range of values.

True or False? The probability of a continuous random variable being exactly equal to a specific value is always zero .

True. The probability of a continuous random variable being exactly equal to a specific value is always zero .

True or False? Increasing the variance of a normal distribution makes its graph taller and narrower .

False. Increasing the variance of a normal distribution makes its graph wider and shorter .

What percentage of data lies within one standard deviation of the mean in a normal distribution?

Approximately 68% of the data lies within one standard deviation of the mean in a normal distribution.

True or False? Nearly all the data lies within three standard deviations of the mean in a normal distribution.

True. Nearly all of the data ( 99.7% ) lies within three standard deviations of the mean in a normal distribution.

True or False? The normal distribution is asymmetrical .

False. The normal distribution is symmetrical about its mean.

What is the relationship between mean, median, and mode in a normal distribution?

In a normal distribution, the mean, median, and mode are all equal.

What is the standard normal distribution ?

The standard normal distribution is a normal distribution where the mean is 0 and the standard deviation is 1 .

True or False? Any normal distribution curve can be transformed to the standard normal distribution curve.

True. Any normal distribution curve can be transformed to the standard normal distribution curve.

A z -value is a standardised value that tells you how many standard deviations a value is away from the mean in a normal distribution.

True or False? A negative z -value indicates that the x -value is greater than the mean.

False. A negative z -value indicates that the x -value is less than the mean.

Why are z -values useful?

z -values are useful because they allow us to compare values from different normal distributions .

True or False? The inverse normal distribution function can be used to find the z -value for a given probability .

True. The inverse normal distribution function can be used to find the z -value for a given probability .

IBMathsDPAnalysis & Approaches (AA)SLFlashcards4. Statistics & ProbabilityNormal Distribution

Normal Distribution (DP IB Analysis & Approaches (AA))

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What is a continuous random variable?
A continuous random variable is a random variable that can take any value within a range of values.
True or False?
The probability of a continuous random variable being exactly equal to a specific value is always zero.
True.
The probability of a continuous random variable being exactly equal to a specific value is always zero.
What does the notation $X ~ N (μ, σ^{2})$ mean?
$X ~ N (μ, σ^{2})$ means that the random variable $X$ follows a normal distribution with mean $μ$ and variance $σ^{2}$ .
True or False?
The random variable $X ~ N (330, 16)$ has a standard deviation of 16.
False.
The random variable $X ~ N (330, 16)$ has a standard deviation of 4.
The 16 is the variance of the distribution, $σ^{2}$ . To find the standard deviation, $σ$ , you have to take the square root: $\sqrt{16} = 4$ .
What is the shape of the graph for a normal distribution?
The graph for a normal distribution is bell-shaped.
True or False?
Increasing the variance of a normal distribution makes its graph taller and narrower.
False.
Increasing the variance of a normal distribution makes its graph wider and shorter.
What percentage of data lies within one standard deviation of the mean in a normal distribution?
Approximately 68% of the data lies within one standard deviation of the mean in a normal distribution.
What interval contains approximately 95% of the data in a normal distribution?
The interval $μ \pm 2 σ$ contains approximately 95% of the data in a normal distribution.
I.e., approximately 95% of the data is within two standard deviations of the mean.
True or False?
Nearly all the data lies within three standard deviations of the mean in a normal distribution.
True.
Nearly all of the data (99.7%) lies within three standard deviations of the mean in a normal distribution.
True or False?
The normal distribution is asymmetrical.
False.
The normal distribution is symmetrical about its mean.
What is the relationship between mean, median, and mode in a normal distribution?
In a normal distribution, the mean, median, and mode are all equal.
True or False?
$P (a < X < b)$ is equal to $P (a \leq X \leq b)$ in a normal distribution.
True.
$P (a < X < b)$ is equal to $P (a \leq X \leq b)$ in a normal distribution.
For continuous probability distributions, strict inequalities ( $<, >$ ) and the equivalent weak inequalities ( $\leq, \geq$ ) are interchangeable in probabilities.
If $X$ is a normal variable, what is $P (X = 3)$ equal to?
If $X$ is a normal variable, $P (X = 3) = 0$ .
The probability for any single value in a normal distribution is always zero.
If $X$ is a normal variable, what four pieces of information would you need to enter into your calculator to find a probability of the form $P (a \leq X \leq b)$ ?
To find a probability of the form $P (a \leq X \leq b)$ in your calculator, you would need to enter:
- the value of $μ$ (the mean for the distribution)
- the value of $σ$ (the standard deviation for the distribution)
- the value of $a$
- the value of $b$
Remember to use the standard deviation ( $σ$ ) and not the variance ( $σ^{2}$ )!
If $X$ is a normal random variable, what upper bound should you use to calculate $P (X > a)$ using a calculator?
If $X$ is a normal random variable, then to calculate $P (X > a)$ using a calculator you should select a very large number as the upper bound.
E.g. $9999999999$ or $1 \times 10^{99}$ .
Make sure that the upper bound you use is at least 4 standard deviations above the mean.
If $X$ is a normal random variable, what lower bound should you use to calculate $P (X < a)$ using a calculator?
If $X$ is a normal random variable, then to calculate $P (X < a)$ using a calculator you should select a very large negative number as the lower bound.
E.g. $- 9999999999$ or $- 1 \times 10^{99}$ .
Make sure that the lower bound you use is at least 4 standard deviations below the mean.
True or False?
$P (X < μ)$ is always equal to 0.25 in a normal distribution.
False.
$P (X < μ)$ is always equal to 0.5 in a normal distribution.
For a normal distribution, $P (X < μ) = P (X \leq μ) = P (X > μ) = P (X \geq μ) = 0.5$
For a normal distribution, state the equation for $P (X > a)$ in terms of $P (X < a)$ .
For a normal distribution $P (X > a) = 1 - P (X < a)$
What is the inverse normal distribution function on a GDC used for?
The inverse normal distribution function on a GDC is used to find the bound that gives a particular probability.
E.g. the value of $a$ that gives $P (X < a) = 0.65$ or $P (X > a) = 0.1$ .
What is a quick way to check if your answer makes sense when using the inverse normal distribution function on a GDC?
To check if your answer makes sense when using the inverse normal distribution function on a GDC, verify that
- if $P (X < a)$ is less than 0.5, then $a$ is smaller than the mean
- if $P (X < a)$ is more than 0.5, then $a$ is larger than the mean
- if $P (X > a)$ is less than 0.5, then $a$ is larger than the mean
- if $P (X > a)$ is more than 0.5, then $a$ is smaller than the mean
What is the standard normal distribution?
The standard normal distribution is a normal distribution where the mean is 0 and the standard deviation is 1.
What is $Z ~ N (0, 1^{2})$ the notation for?
$Z ~ N (0, 1^{2})$ is the notation for the standard normal distribution.
True or False?
Any normal distribution curve can be transformed to the standard normal distribution curve.
True.
Any normal distribution curve can be transformed to the standard normal distribution curve.
What is a z-value?
A z-value is a standardised value that tells you how many standard deviations a value is away from the mean in a normal distribution.
True or False?
A negative z-value indicates that the x-value is greater than the mean.
False.
A negative z-value indicates that the x-value is less than the mean.
State the formula for calculating a z-value from an x-value, in terms of the mean, $μ$ and standard deviation, $σ$ .
If $X ~ N (μ, σ^{2})$ and $Z ~ N (0, 1^{2})$ , then the formula for calculating a z-value from an x-value is $z = \frac{x - μ}{σ}$ .
Where:
- $μ$ is the mean of the distribution of the variable $X$
- $σ$ is the standard deviation of the distribution of the variable $X$
This formula is in the exam formula booklet.
Why are z-values useful?
z-values are useful because they allow us to compare values from different normal distributions.
What is a good first step in finding an unknown parameter (mean, $μ$ , or standard deviation, $σ$ ) in a normal distribution?
A good first step in finding an unknown parameter (mean, $μ$ , or standard deviation, $σ$ ) in a normal distribution is to sketch the normal curve.
True or False?
The inverse normal distribution function can be used to find the z-value for a given probability.
True.
The inverse normal distribution function can be used to find the z-value for a given probability.
How many probabilities are needed to find the mean, $μ$ and standard deviation, $σ$ when both parameters are unknown?
Two probabilities are needed to find the mean, $μ$ and standard deviation, $σ$ when both parameters are unknown.

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