Suppose a population is known to be normally distributed with a mean, µ, equal to 116 and a standard deviation, o, equal to 14. Approximately what percent of the population would be between 116 and 130?
Suppose a population is known to be normally distributed with a mean, µ, equal to 116 and a standard deviation, o, equal to 14. Approximately what percent of the population would be between 116 and 130?
Suppose a population is known to be normally distributed with a mean, µ, equal to 116 and a standard deviation, o, equal to 14. Approximately what percent of the population would be between 116 and 130?
Suppose a population is known to be normally distributed with a mean, μ, equal to 116 and a standard deviation, σ, equal to 14. Approximately what percent of the population would be between 116 and 130?
Transcribed Image Text:**Question: Normal Distribution Analysis**
Suppose a population is known to be normally distributed with a mean, \( \mu \), equal to 116 and a standard deviation, \( \sigma \), equal to 14. Approximately what percent of the population would be between 116 and 130?
**Answer Options:**
- 47.5%
- 81.5%
- 13.5%
- 95%
- 34%
- 68%
**Explanation:**
In a normal distribution, data falls symmetrically around the mean, creating a bell-shaped curve. The empirical rule (68-95-99.7 rule) is often used to estimate the data spread:
- **68%** of values fall within one standard deviation (\( \sigma \)) of the mean.
- **95%** fall within two standard deviations.
- **99.7%** fall within three standard deviations.
To find the percentage of the population between 116 and 130:
1. Calculate the number of standard deviations between 116 and 130.
2. \( 130 - 116 = 14 \), which is one standard deviation from the mean.
Thus, approximately **34%** of the population is expected to be between the mean and one standard deviation above the mean (half of 68% because the distribution is symmetrical).
Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...
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