Which of the following statements are TRUE about the Normal Distribution? Check all that apply.
a. 50% of the data values lie at or above the mean.
b. A z-score is the number of standard deviations a specific data value is from the mean of the distribution.
c.The graph of the Normal Distribution is bell-shaped, with tapering tails that never actually touch the horizontal axis.
d. Data values farther from the mean are less common than data values closer to the mean.
e. The mean corresponds to the z-score of 1.
f. About 95% of all data values lie within 1 standard deviation of the mean.
g. The Empirical Rule only applies when a value is exactly 1, 2, or 3 standard deviations away from the mean.
h. Data values are spread evenly around the mean.
I. The distribution is symmetric with a single peak.
J. The mean, median and mode are all equal and occur at the center of the distribution.
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David Lane normal distribution
The true statements about the Normal Distribution are:
b. A z-score is the number of standard deviations a specific data value is from the mean of the distribution.
c. The graph of the Normal Distribution is bell-shaped, with tapering tails that never actually touch the horizontal axis.
d. Data values farther from the mean are less common than data values closer to the mean.
f. About 95% of all data values lie within 1 standard deviation of the mean.
h. Data values are spread evenly around the mean.
i. The distribution is symmetric with a single peak.
j. The mean, median, and mode are all equal and occur at the center of the distribution.
To determine which statements are true about the Normal Distribution, let's go through each statement and explain how to find the answer:
a. 50% of the data values lie at or above the mean.
To determine if this statement is true, we need to understand the concept of a normal distribution. In a normal distribution, the mean is at the center of the distribution, and the distribution is symmetric. Therefore, 50% of the data values will lie at or above the mean. So, this statement is TRUE.
b. A z-score is the number of standard deviations a specific data value is from the mean of the distribution.
A z-score measures the number of standard deviations a specific data value is away from the mean of a distribution. So, this statement is TRUE.
c. The graph of the Normal Distribution is bell-shaped, with tapering tails that never actually touch the horizontal axis.
The Normal Distribution is characterized by a bell-shaped curve that smoothly tapers at the tails. The tails of the distribution never touch the horizontal axis. So, this statement is TRUE.
d. Data values farther from the mean are less common than data values closer to the mean.
In a normal distribution, data values closer to the mean are more common, while data values farther away from the mean become less frequent. So, this statement is TRUE.
e. The mean corresponds to the z-score of 1.
The mean of a normal distribution always has a z-score of 0, indicating that it is exactly at the center. Therefore, this statement is NOT TRUE.
f. About 95% of all data values lie within 1 standard deviation of the mean.
According to the Empirical Rule, approximately 68% of data values lie within one standard deviation of the mean in a normal distribution. Therefore, this statement is NOT TRUE.
g. The Empirical Rule only applies when a value is exactly 1, 2, or 3 standard deviations away from the mean.
The Empirical Rule is a rule of thumb that applies to normal distributions. It states that approximately 68%, 95%, and 99.7% of the data falls within 1, 2, and 3 standard deviations of the mean, respectively. Therefore, this statement is NOT TRUE.
h. Data values are spread evenly around the mean.
In a normal distribution, data values gradually decrease in frequency as they move away from the mean. So, this statement is NOT TRUE.
i. The distribution is symmetric with a single peak.
A key characteristic of the Normal Distribution is that it is symmetric, meaning the left and right sides of the distribution are mirror images of each other. Additionally, it has a single peak, which is the mean. Therefore, this statement is TRUE.
j. The mean, median, and mode are all equal and occur at the center of the distribution.
In a perfectly symmetrical normal distribution, the mean, median, and mode are indeed equal and occur at the center of the distribution. However, this is not always the case in all distributions. Therefore, this statement could be TRUE, but it is not always true for all distributions.
Based on the explanations provided, the TRUE statements about the Normal Distribution are: a, b, c, d, i.