True or false: a. All normal distributions are symmetrical b. All normal distributions have a mean of 1.0 c. All normal distributions have a standard deviation of 1.0 d. The total area under the curve of all normal distributions is equal to 1

necessaryh

necessaryh

Answered question

2020-12-01

True or false:
a. All normal distributions are symmetrical
b. All normal distributions have a mean of 1.0
c. All normal distributions have a standard deviation of 1.0
d. The total area under the curve of all normal distributions is equal to 1

Answer & Explanation

liannemdh

liannemdh

Skilled2020-12-02Added 106 answers

Step 1
a) True, all normal distributions are symmetrical, bell shaped curve.
Step 2
b) False: Not all normal distributions have mean as 1. Means vary depending on the type of data.
Step 3
c) False: Not all normal distributions have standard deviations as 1. Standard deviation, or for that matter, variance vary depending on the type of data.
Step 4
d) True: Total area under the curve of all normal distribution is 1.
karton

karton

Expert2023-06-19Added 613 answers

a. False. Not all normal distributions are symmetrical. A normal distribution can be symmetrical, but it can also be skewed to the left or right.
b. False. All normal distributions do not have a mean of 1.0. The mean of a normal distribution can take any real value.
c. False. All normal distributions do not have a standard deviation of 1.0. The standard deviation of a normal distribution can take any positive value.
d. True. The total area under the curve of all normal distributions is equal to 1. The area under the entire normal distribution curve represents the probability of the random variable taking any value within its range, and the total probability must sum to 1.
star233

star233

Skilled2023-06-19Added 403 answers

a. True: All normal distributions are symmetrical.
A normal distribution is a bell-shaped curve that is symmetric around its mean. It has equal probabilities on both sides of the mean, resulting in a symmetrical distribution. This symmetry means that the left and right tails of the distribution mirror each other. Mathematically, a normal distribution is characterized by its mean (μ) and standard deviation (σ), and the symmetry holds for all normal distributions. Therefore, the statement is true.
b. False: Not all normal distributions have a mean of 1.0.
The mean of a normal distribution can take any real value. It is a measure of central tendency that represents the average value of the distribution. While some normal distributions may have a mean of 1.0, there are many other possible values for the mean. It depends on the specific distribution and the data it represents. Therefore, the statement is false.
c. False: Not all normal distributions have a standard deviation of 1.0.
The standard deviation of a normal distribution measures the spread or dispersion of the data around the mean. It quantifies how much the values in the distribution deviate from the mean. While a standard deviation of 1.0 is a common choice and often used as a reference point, it is not a requirement for all normal distributions. Different normal distributions can have varying standard deviations, indicating different levels of variability in the data. Therefore, the statement is false.
d. True: The total area under the curve of all normal distributions is equal to 1.
The total area under the curve of a normal distribution is always equal to 1. This property is known as the normalization property and is a fundamental characteristic of a probability distribution. The area under the curve represents the probability of the random variable falling within a certain range of values. Since the probability of the entire sample space (all possible values) is 1, the total area under the curve of a normal distribution must also be 1. Therefore, the statement is true.
alenahelenash

alenahelenash

Expert2023-06-19Added 556 answers

Result:
a) False
b) False
c) False
d) True
Solution:
a. False. Not all normal distributions are symmetrical. While the standard bell-shaped normal distribution is symmetrical, there are also skewed normal distributions that are not symmetrical.
b. False. Not all normal distributions have a mean of 1.0. The mean of a normal distribution can take any real value. The normal distribution is characterized by its mean and standard deviation, but the mean value is not fixed at 1.0.
c. False. Not all normal distributions have a standard deviation of 1.0. The standard deviation of a normal distribution can also take any positive real value. The standard deviation determines the spread or variability of the data, but it is not fixed at 1.0 for all normal distributions.
d. True. The total area under the curve of all normal distributions is equal to 1. This property is known as the normalization condition. The area under the entire curve represents the probability of observing any value within the distribution, and it must sum up to 1.

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