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Chapter 17: Problem 11

X has a normal distribution with mean \(91 .\) Given that the probability that \(x\) is larger than 104 is \(24.6 \%,\) find the standard deviation \(\sigma\) of this distribution.

Short Answer

Expert verified
The standard deviation is approximately 18.98.

Step by step solution

01

Understand the Problem

We are given a normal distribution with a mean \( \mu = 91 \). We need to find the standard deviation \( \sigma \) given that the probability \( P(X > 104) = 24.6\% \).
02

Convert Probability to z-score

To solve this problem, first, we need to convert the probability \( P(X > 104) = 0.246 \) to a z-score using the standard normal distribution table. This corresponds to finding the z-score for the cumulative probability \( P(Z < z) = 1 - 0.246 = 0.754 \).
03

Find the z-score

Using the standard normal distribution table, we find the z-score corresponding to \( P(Z < z) = 0.754 \). The z-score approximately corresponding to this cumulative probability is \( z = 0.685 \).
04

Use the z-score Formula

We use the formula for the z-score: \[ z = \frac{X - \mu}{\sigma} \]where \( X = 104 \), \( \mu = 91 \), and \( z = 0.685 \).
05

Solve for \( \sigma \)

Plugging the values into the z-score formula, we have:\[ 0.685 = \frac{104 - 91}{\sigma} \]Simplifying this, we get:\[ \sigma = \frac{13}{0.685} \approx 18.98 \]
06

Conclusion

The standard deviation \( \sigma \) of the distribution is approximately \(18.98\).

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Key Concepts

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

Understanding Standard Deviation
The standard deviation (\(\sigma\)) is a measure of how spread out the numbers are in a data set. In a normal distribution, it's especially important because it helps determine the shape of the data around the mean.Let's break it down:
  • Measure of Spread: Standard deviation indicates how much variation there is from the average (mean). A low standard deviation means that the data points tend to be close to the mean. A high standard deviation indicates that the data points are spread out over a larger range of values.
  • In Context: For example, if we know the mean is 91 and the standard deviation is 18.98, data points are typically spread within a range around 91.
In the context of our exercise, we used the z-score relation to find \(\sigma\) once we had the mean and a particular outcome, demonstrating the practical use of standard deviation in understanding real data distributions.
Demystifying the Z-Score
A z-score is a value that tells you how many standard deviations a particular data point is from the mean. It's a key concept in statistics used to understand where a data point stands in a distribution.Here are some crucial points about z-scores:
  • Standardization: Z-scores allow for the standardization of data, making it easier to compare different data points. Regardless of the raw data value's unit, z-scores provide a context-free comparison measure.
  • Calculation: It's calculated using the formula: \( z = \frac{X - \mu}{\sigma} \), where \(X\) is the raw score, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.
When dealing with the normal distribution in our problem, we used the z-score to convert the given probability into a usable value. This way, we were able to relate 104 (raw score) to the mean, determining how far off it was from our distribution's average.
The Role of the Mean in Distributions
The mean (\(\mu\)) is the average of all data points in a set and a central value in statistics.Some key insights about the mean are:
  • Central Tendency: It gives us an idea of the central tendency of the data set. All data points contribute to this calculation, ensuring it provides a fair representation.
  • In Normal Distribution: The mean is at the center of a normal distribution curve. The shape of the normal curve is symmetrical around this mean value.
In our exercise, you'll see that the mean acts as a reference point. Knowing the mean allowed us to determine how far 104 deviated from what is typically expected, utilizing the standard deviation context, leading to the computation of a z-score to understand this deviation quantitatively.

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