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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 ii. \(95 \%\) \(\begin{array}{ll}\text { iii. } 68 \% & \text { iv. } 50 \%\end{array}\) b. Roughly what percentage of \(z\) -scores are between \(-3\) and 3 ? \(\begin{array}{llll}\text { i. almost all } & \text { ii. } 95 \% & \text { uii. } 68 \% & \text { iv. } 50 \%\end{array}\) c. Roughly what percentage of \(z\) -scores are between \(-1\) and 1 . i. almost all ii. \(95 \%\) iii. \(68 \%\) iv. \(50 \%\) d. Roughly what percentage of \(z\) -scores are greater than 0 ? i. almost all ii. \(95 \%\) iii. \(68 \%\) iv. \(50 \%\) e. Roughly what percentage of \(z\) -scores are between 1 and 2 ? \(\begin{array}{llll}\text { i. almost all } & \text { ii. } 13.5 \% & \text { uii. 50\% iv. } 2 \%\end{array}\)

Step by step solution

01

Apply Empirical Rule for \(z\)-scores between -2 and 2

According to the empirical rule, about 95% of the data in a normal distribution is within 2 standard deviations from the mean. Therefore, the answer is ii. 95%.

02

Apply Empirical Rule for \(z\)-scores between -3 and 3

Similarly, for data within 3 standard deviations from the mean, about 99.7% of the data fits this range. Thus, choosing the closest option, the answer is i. Almost all (99.7% rounds to nearly 100%)

03

Apply Empirical Rule for \(z\)-scores between -1 and 1

In this case, about 68% of the data in a normal distribution is within 1 standard deviation from the mean. Therefore, the answer is iii. 68%.

04

Apply Empirical Rule for \(z\)-scores greater than 0

Noting that the standard normal distribution is symmetric about 0, meaning exactly half of all data sample would be greater than 0 and the other half less than 0 in a standard normal distribution, answer would be iv. 50%.

05

Apply Empirical Rule for \(z\)-scores between 1 and 2

In a normal distribution about 68% of data is within 1 standard deviation from the mean and about 95% of the data is within 2 standard deviations. By subtracting the percentage of data within 1 standard deviation from the percentage within 2 standard deviations. i.e. 95% - 68% = 27%. However, this 27% applies to both sides of the distribution (between -2 and -1, and between 1 and 2). So for data between 1 standard deviation and 2 standard deviations to either side of the mean, the answer is ii. 13.5%.

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Key Concepts

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

Normal Distribution

The normal distribution is a very important concept in statistics and probability. Often called the "bell curve," it represents how data points are distributed around the mean, which is the average value. Most data points are clustered around the mean, with fewer points appearing as you move further away toward the extremes. This symmetric shape is why it’s also referred to as a Gaussian distribution.

In a perfect normal distribution, the mean, median, and mode are all the same and are located at the center of the distribution. This means that the data set is balanced and mirrored on either side of the mean. Understanding the normal distribution is crucial, as it forms the basis for the Empirical Rule and many statistical calculations.

• Symmetric around the mean
• Mean = Median = Mode
• Forms a bell-shaped curve

Standard Deviations

Standard deviations measure the spread of data points in a data set. Specifically, it tells us how much the data deviates from the mean. A lower standard deviation indicates that the data points tend to be very close to the mean, while a higher standard deviation indicates a wider spread.

In the context of a normal distribution, standard deviations help break the distribution into predictable segments. According to the Empirical Rule for normal distributions, approximately 68% of data points fall within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three. Thus, understanding standard deviations allows us to make informed predictions about where most data points will lie.

• Quantifies the variation in data
• Helps define the Empirical Rule
• Essential for understanding data distribution

Z-scores

Z-scores are a statistical measurement that describes a value's position relative to the mean of a group of values. They are expressed in terms of standard deviations and provide insight into how far away a particular data point is from the mean. This makes it easier to understand and compare different data points, even if they belong to different distributions.

For example, a z-score of 0 indicates that the data point is exactly at the mean, while a z-score of +2 or -2 indicates that the data point is two standard deviations away from the mean. This standardization allows for easy comparison across different datasets by putting them on a common scale.

• Measures position relative to the mean
• Expressed in standard deviations
• Allows comparison across different datasets

Probability Approximation

Probability approximation involves estimating the likelihood of a certain event occurring within a statistical context. Within normal distributions, this approximation is often guided by the Empirical Rule, which gives us an easy way to estimate the probability of data falling within certain ranges of standard deviations.

By employing normal distribution properties, you can predict probabilities without having to compute each one individually. The normal distribution curve allows for quick approximations, such as knowing that about 95% of data lies within two standard deviations of the mean. These estimations are particularly useful in fields such as finance, medicine, and social sciences where making predictions based on historical data patterns is essential.

• Estimates likelihoods within distributions
• Guided by the Empirical Rule
• Useful for practical predictions and decision-making