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Rule with z-Scores The Empirical Rule applies rough approximations to probabilities for any unimodal, symmetric distribution. But for the Normal distribution we can be more precise. Use the figure and the fact that the Normal curve is symmetric to answer the questions. Do not use a Normal table or technology. According to the Empirical Rule, a. Roughly what percentage of \(z\) -scores are between \(-2\) and 2 ? i. almost all iii. \(68 \%\) ii. \(95 \%\) iv. \(50 \%\) b. Roughly what percentage of \(z\) -scores are between \(-3\) and 3 ? i. almost all iii. \(68 \%\) ii. \(95 \%\) iv. \(50 \%\) c. Roughly what percentage of \(z\) -scores are between \(-1\) and 1 . i. almost all iii. \(68 \%\) ii. \(95 \%\) iv. \(50 \%\) d. Roughly what percentage of \(z\) -scores are greater than \(0 ?\) i. almost all iii. \(68 \%\) ii. \(95 \%\) iv. \(50 \%\) e. Roughly what percentage of \(z\) -scores are between 1 and \(2 ?\) i. almost all iii. \(50 \%\) ii. \(13.5 \%\) iv. \(2 \%\)

Step by step solution

01

Calculation of percentage of z-scores between -2 and 2

According to the Empirical Rule, about 95% of the z-scores (data within two standard deviations from the mean) are between -2 and 2.

02

Calculation of percentage of z-scores between -3 and 3

The normal distribution's symmetry and bell-like shape cause 99.7% of z-scores to fall between 3 standard deviations (both negative and positive) of the mean, according to the Empirical Rule. Consequently, almost all z-scores are between -3 and 3.

03

Calculation of percentage of z-scores between -1 and 1

According to the Empirical Rule, about 68% of the z-scores (data within one standard deviation from the mean) are between -1 and 1.

04

Calculation of percentage of z-scores greater than 0

The normal distribution is symmetric around the mean. Therefore, half of the z-scores are greater than 0 while the other half are less than 0, resulting in 50% of z-scores being greater than 0.

05

Calculation of percentage of z-scores between 1 and 2

Approximately 95% of z-scores are within two standard deviations of the mean. And we know that roughly 68% falls within one standard deviation. To find the percentage that falls between one and two standard deviations, we subtract 68% from 95%, and get 27%. As the distribution is symmetric, this 27% must be split evenly around the mean, so the percentage of z-scores between 1 and 2 is about 13.5%.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

z-Scores

Understanding z-scores is essential for statistics students, as it is a way to measure the distance of a data point from the mean when data is standardized. A z-score is expressed in terms of standard deviations from the mean. For example, a z-score of 1 indicates that the data point is one standard deviation above the mean, while a z-score of -1 means it's one standard deviation below the mean.

A z-score can be calculated using the formula:
\[ z = \frac{(X - \mu)}{\sigma} \]
where \(X\) is the data point, \(\mu\) is the mean, and \(\sigma\) is the standard deviation of the dataset. With this standardized score, we can compare data from different sets or distributions easily. In the textbook exercise, percentages of data within certain ranges of z-scores are calculated, illustrating the practical application of z-scores in determining the relative position of data points within a distribution.

Normal Distribution

The Normal distribution, also known as the Gaussian distribution, is a continuous probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. The shape of the normal distribution is often referred to as a 'bell curve.'

An important feature of the normal distribution is that it is completely determined by its mean and standard deviation. The mean of the distribution determines where the center of the curve is located, while the standard deviation determines how spread out the curve is.

In the context of our exercise, the percentage of z-scores that fall between certain intervals is found without the need for a normal distribution table, thanks to the properties of the distribution where we know that about 68%, 95%, and 99.7% of the data fall within one, two, and three standard deviations from the mean, respectively.

Standard Deviation

The standard deviation showcases the amount of variation or dispersion in a set of values. It is a core concept in statistics, representing how spread out the numbers are in a dataset. A low standard deviation means that the data points are generally close to the mean, while a high standard deviation indicates that the data points are spread out over a wider range.

The formula for standard deviation is:
\[ \sigma = \sqrt{\frac{\sum (X - \mu)^2}{n}} \]
where \(\sigma\) is the standard deviation, \(X\) represents each value in the dataset, \(\mu\) is the mean of the data, and \(n\) is the number of data points. In the exercise provided, the empirical rule uses multiples of the standard deviation to approximate the percentage of data within certain ranges. This shows the direct application of standard deviation in understanding data dispersion.

Symmetric Distribution

A symmetric distribution is a type of probability distribution where the left half of the histogram (distribution of values) mirrors the right half. The normal distribution is a perfect example of a symmetric distribution. Symmetry in a distribution means that the mean, median, and mode of the distribution are all equal and are located at the center of the distribution.

In symmetric distributions, probabilities on either side of the mean are the same. This property is used in our exercise to calculate the percentage of z-scores greater than zero and to understand the percentages that fall between different ranges of z-scores. The symmetry allows us to infer that 50% of z-scores are above the mean (greater than 0) and 50% are below, reinforcing the importance of symmetry in statistical analysis.