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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

Answer (a)

For a normal distribution, about \(95\%\) of the z-scores lie between -2 and 2. This is a direct application of the empirical rule.

02

Answer (b)

Again, applying the empirical rule, we know about \(99.7\%\) (or almost all) of the z-scores are between -3 and 3.

03

Answer (c)

For z-scores lying between -1 and 1, the data falls within one standard deviation of the mean. Hence, approximately \(68\%\) of z-scores lie in this range.

04

Answer (d)

Since the normal distribution is symmetric about the mean (which is 0 for z-scores), roughly \(50\%\) of the z-scores are greater than 0.

05

Answer (e)

The percentage of z-scores lying between 1 and 2 can be found indirectly using empirical rule. We know 95% lie between -2 and 2. We subtract the percentage that lies within -1 and 1 is 68%. So, we get \(95 - 68 = 27\%\). But, this percentage refers to scores between -2 and -1 and 1 and 2. Since the distribution is symmetric, the scores between 1 and 2 must be \(27/2 = 13.5\%\).

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Key Concepts

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

Normal Distribution

A normal distribution, often referred to as the Gaussian distribution, describes a continuous probability distribution that is symmetric around its mean. You can think of it as a bell-shaped curve, where the majority of observations cluster around the central peak.

This distribution applies to many real-world phenomena, such as heights, test scores, or measurement errors.

• Centered around the mean: The highest point of the curve is at the mean, which is also the median and mode.
• Symmetrical: If you were to fold the curve in half at the mean, both sides would align perfectly.
• Defined by mean and standard deviation: The mean determines the center, while the standard deviation dictates the spread or width of the curve.

The Empirical Rule, in particular, relies on this symmetry and the consistent shape of the curve to make quick and useful approximations about data.

Z-scores

Z-scores are a way to quantify how far away a particular data point is from the mean, in terms of standard deviations.

• If a data point has a z-score of 0, it is exactly the mean.
• A positive z-score indicates a value above the mean, while a negative z-score shows it is below the mean.

To calculate a z-score, use the formula \[ z = \frac{x - \mu}{\sigma}\]where

• \(x\) is the data point in question,
• \(\mu\) is the mean of the dataset, and
• \(\sigma\) is the standard deviation.

This concept is crucial in understanding the percentage distributions given by the Empirical Rule, as it translates those distributions into various standard deviations.

Probability Approximations

Probability approximations are key when using the Empirical Rule to make educated guesses about data lying within certain ranges of a normal distribution.

These approximations allow us to understand the likelihood of a data point falling within a specific interval:

• Approximately \(68\%\) of data lies between one standard deviation from the mean (i.e., \(-1\) to \(1\) z-scores).
• About \(95\%\) falls within two standard deviations (i.e., \(-2\) to \(2\) z-scores).
• Almost all, \(99.7\%\), reside between three standard deviations (i.e., \(-3\) to \(3\) z-scores).

Through these probability approximations, the Empirical Rule becomes a powerful tool for quickly gauging how much data falls within these intervals, especially without turning to more complex statistical methods.

Symmetry in Distributions

Symmetry in distributions is a distinctive feature of normal distributions that makes them mathematically convenient and widely applicable.

Because of symmetry, characteristics on one side of the mean are mirrored on the other:

• A symmetric distribution implies identical probabilities on both sides of the mean.
• This symmetry helps in simplifying calculations, as demonstrated by ease in predicting probabilities with z-scores.
• For any given z-score distance from the mean, the probability on the negative side mirrors that on the positive side.

Understanding symmetry's role in distributions elevates its importance beyond aesthetics, proving essential for both foundational statistical principles and practical applications, such as applying the Empirical Rule.