40 Questions
Did you know that in 1733, Abraham de Moivre first discovered the ______ distribution.
normal
The French mathematician De Moivre introduced the ______ probability distribution.
normal
The normal curve is often called the ______ Distribution.
Gaussian
The normal distribution provides a graphical representation of statistical values that are needed in describing the characteristics of ______ as well as in making decisions.
population
A random variable X is said to be normally distributed with mean µ and standard ______.
deviation
The distribution curve is ______-shaped, and has a single peak.
bell
The mean, median and mode coincide at the ______.
center
The width of the curve is determined by the standard ______ of the distribution.
deviation
The z – score is a measure of relative standing that tells how many _______________________ deviations either above or below the mean a particular value is.
standard
The scores represent the distances from the center measured in _______________________ deviation units.
standard
There are six z-score values at the baseline of the normal curve: three z-scores to the left of the mean that are _______________________ and three z-scores to the right of the mean which are positive.
negative
Raw scores may be composed of large values, but these large values cannot be accommodated at the baseline of the normal curve, so these need to be transformed into scores for convenience without sacrificing meanings associated with the raw _______________________.
scores
The areas under the normal curve are given in terms of z values or _______________________.
scores
Either the z-scores locate X within a _______________________ or within a population.
sample
The raw score X is above the mean if z is _______________________ and it is below the mean when z is negative.
positive
Find the z-score value that corresponds to a normal random variable X = 93 in the given mean µ = 85 and standard deviation σ = 8 of a _______________________ in Statistics test.
population
To find the ______ of the regions under the normal curve, simply find the area of the given z-value using the z-Table.
areas
The z-Table is also known as the ______ of Areas under the Normal Curve.
Table
To find the area that corresponds to a z-score, express the given value into ______ decimal form.
two
In the z-Table, find the Row with the z-value and the Column with the heading ______.
.00
The z-score is also known as the ______ score.
standard
To find the area under the standard normal curve, sketch the area between the given z-scores and find the area using the ______.
z-Table
The z-score can be found in the ______ of the normal curve.
regions
The z-score is a measure of how many ______ units an observation is away from the mean.
standard
To find the probability of the area below z = 0.50, use the notation P(z __________ a).
less than
To find the probability of the area at least z = -2, use the notation P(z __________ a).
greater than
The notation P(z < a) is used to find the probability of the area __________ z = a.
below
The notation P(z > a) is used to find the probability of the area __________ z = a.
above
To find the probability of the area between z = -1.5 and z = 2, use the notation P(__________ < z < b).
a
The probability of the area below z = 0.50 is __________ or 69.15%.
0.6915
The probability of the area at least z = -2 is __________ or 97.72%.
0.9772
The probability notation P(z > a) is equal to 1 - P(z __________ a).
less than
The standard score is also known as ______ score.
z
The probability of a normal distribution is a number from 0 to ______.
1
The z-Table is also known as the ______ of Areas under the Normal Curve.
Table
The notation P ( z < a ) denotes the probability that the z-score is less than ______.
a
The probability notation P ( a < z < b ) denotes the probability that the z-score is between ______ and ______.
a and b
The z-score value is used to find the ______ of Matt in Statistics and Probability.
standardized score
The normal distribution is used to describe the characteristics of ______ as well as in making decisions.
statistical values
The probability of a normal distribution can be shown as ______ under the standard normal curve.
areas
Study Notes
Normal Distribution
- The normal distribution, also known as the Gaussian Distribution, was first discovered by Abraham de Moivre in 1733.
- It is a bell-shaped distribution that plays a crucial role in inferential statistics.
- A normal random variable X is said to be normally distributed with mean µ and standard deviation σ.
Properties of Normal Distribution
- The distribution curve is bell-shaped and has a single peak, making it unimodal.
- The curve is symmetrical about its center.
- The mean, median, and mode coincide at the center.
- The width of the curve is determined by the standard deviation of the distribution.
Areas Under the Normal Curve
- To find the areas of the regions under the normal curve, use the z-Table (Table of Areas under the Normal Curve).
- Example: Find the area that corresponds to a z-score value of 0.6 by finding the intersection of the row z = 0.6 and the column with the heading .00 in the z-Table.
Standard Score (Z-Score)
- A z-score is a measure of relative standing that tells how many standard deviations either above or below the mean a particular value is.
- The scores represent the distances from the center measured in standard deviation units.
- There are six z-score values at the baseline of the normal curve: three z-scores to the left of the mean (negative) and three z-scores to the right of the mean (positive).
- The importance of z-score lies in its ability to transform raw scores into scores for convenience without sacrificing meanings associated with the raw scores.
Formula for Z-Score
- z = (X - µ) / σ
- where X is the given measurement of a normal random variable, µ is the population mean, σ is the population standard deviation, 𝑥̅ is the sample mean, and s is the sample standard deviation.
Applications of Normal Distribution
- The probability, or proportion, or the percentage associated with the specific sets of measurement values can be found using the normal distribution.
- All probabilities associated with the standard normal random variables can be shown as areas under the standard normal curve.
- The probability notation P (z < a) denotes the probability that the z-score is less than a, P (z > a) denotes the probability that the z-score is greater than a, and P (a < z < b) denotes the probability that the z-score is between a and b.
Examples of Applications of Normal Distribution
- Example 1: Find the probability of the area below z = 0.50 by consulting the z-Table and finding the area that corresponds to z = 0.50.
- Example 2: Find the area that is at least z = -2 by consulting the z-Table and finding the area that corresponds to z = -2.00.
- Example 3: Find the area between z = -1.5 and z = 2 by consulting the z-Table and finding the area that corresponds to z = 2 and then subtracting the area that corresponds to z = -1.5.
Test your understanding of normal distribution concepts, including areas under the normal curve and standard scores. Learn about the history and applications of normal distribution.
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