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Chapter 6: Problem 35

Suppose X ~ N(鈥�3, 1). Between what x values does 95.45% of the data lie? The range of x values is centered at the mean of the distribution(i.e., 鈥�3).

Short Answer

Expert verified
The range is between \(-5\) and \(-1\).

Step by step solution

01

Understand the Normal Distribution

We have a normal distribution, denoted as \(X \sim N(\mu, \sigma)\), where \(\mu\) is the mean and \(\sigma\) is the standard deviation. Here, \(\mu = -3\) and \(\sigma = 1\). The question asks for the range of x values that encompasses 95.45% of the data, centered at the mean.
02

Recall Empirical Rule

The empirical rule for normal distributions states that approximately 68.27% of data lies within one standard deviation (\(\sigma\)), 95.45% within two standard deviations, and 99.73% within three standard deviations of the mean. We use the 95.45% interval for this exercise.
03

Calculate Range using the Empirical Rule

Since 95.45% of the data lies within two standard deviations from the mean, we calculate this interval as \(\mu - 2\sigma\) to \(\mu + 2\sigma\). This gives us the interval \(-3 - 2(1)\) to \(-3 + 2(1)\).
04

Simplify and State the Range

Simplifying the range from the previous step, we calculate \(-3 - 2 = -5\) and \(-3 + 2 = -1\). Therefore, 95.45% of the data lies between \(-5\) and \(-1\).

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Key Concepts

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

Understanding the Empirical Rule
The empirical rule is a handy guideline in statistics that helps us understand how data spreads in a normal distribution. When we talk about a normal distribution, we're referring to a bell-shaped curve where most of the data clusters around the central point, or the mean.
This rule simplifies the understanding of how data points are distributed on the bell curve:
  • 68.27% of data falls within one standard deviation from the mean.
  • 95.45% of data falls within two standard deviations.
  • 99.73% falls within three standard deviations.
In our exercise, we specifically focus on the range where 95.45% of the data points lie. This can be found within two standard deviations from the mean. Therefore, the empirical rule helps in predicting that most of the data, specifically 95.45%, will fall between the calculated values when you know the mean and standard deviation.
Diving into Standard Deviation
Standard deviation is a crucial statistic that measures how much data varies around the mean or average value. It tells us how spread out the values in a data set are. In a normal distribution, understanding standard deviation helps determine where the majority of data points lie relative to the mean.
The standard deviation is denoted by the symbol \(\sigma\), and in this exercise, it is given as 1. This means that most data points are tightly packed around the mean. When using the empirical rule:
  • A standard deviation of 1 indicates that the central 68.27% of data is between -4 and -2.
  • Furthermore, the central 95.45% will extend from -5 to -1.
Thus, standard deviation is key in calculating the intervals around the mean, giving us insights into the data's spread and concentration.
The Role of Mean in Statistics
The mean is a central concept in statistics, representing the average value of a data set. In a normal distribution, the mean is the peak of the bell curve, where data is most densely clustered.
For this exercise, the mean is -3, denoting the center point of our data's distribution. The mean's position directly informs us where the central point of our calculated intervals will be. In a standard normal distribution, calculations using the empirical rule rely heavily on the mean.
Calculating the range in which 95.45% of data lies involves using the mean to start at the center (at -3) and then extending to either side using the standard deviation:
  • Moving two standard deviations to the left and right from the mean, we find our range between -5 and -1.
The mean thus anchors our calculation of where the majority of data is expected, providing a critical reference point in statistical analysis.

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Most popular questions from this chapter

\(X \sim N(6,2)\) Find the probability that \(x\) is between three and nine.

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