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Magazine Chapter 5: Problem 25
Let \(X\) be a random variable normally distributed with parameters \(\mu=70\),
\(\sigma=10 .\) Estimate
(a) \(P(X>50)\).
(b) \(P(X<60)\).
(c) \(P(X>90)\)
(d) \(P(60
Short Answer
Expert verified
(a) \(0.9772\), (b) \(0.1587\), (c) \(0.0228\), (d) \(0.6826\).
Step by step solution
01
Understanding the Normal Distribution
The random variable \(X\) is normally distributed with a mean (\(\mu\)) of 70 and a standard deviation (\(\sigma\)) of 10. We will use the standard normal distribution, mean 0 and standard deviation 1, by standardizing \(X\) into a \(Z\)-score.
02
Calculating the Z-score Formula
The Z-score formula is given by: \[Z = \frac{X - \mu}{\sigma}\]where \(\mu = 70\) and \(\sigma = 10\). We will use this formula to find \(Z\)-scores for the values of \(X\) to be used in the standard normal distribution.
03
Calculating P(X > 50)
First, calculate the \(Z\)-score for \(X = 50\):\[Z = \frac{50 - 70}{10} = -2\]Using the standard normal distribution table, \(P(Z > -2) = 1 - P(Z < -2) \approx 0.9772\). Thus, \(P(X > 50) \approx 0.9772\).
04
Calculating P(X < 60)
Calculate the \(Z\)-score for \(X = 60\):\[Z = \frac{60 - 70}{10} = -1\]Using the standard normal distribution table, \(P(Z < -1) \approx 0.1587\). Thus, \(P(X < 60) \approx 0.1587\).
05
Calculating P(X > 90)
Calculate the \(Z\)-score for \(X = 90\):\[Z = \frac{90 - 70}{10} = 2\]Using the standard normal distribution table, \(P(Z > 2) = 1 - P(Z < 2) \approx 0.0228\). Thus, \(P(X > 90) \approx 0.0228\).
06
Calculating P(60 < X < 80)
Calculate the \(Z\)-scores for \(X = 60\) and \(X = 80\):\[Z_{60} = \frac{60 - 70}{10} = -1\]\[Z_{80} = \frac{80 - 70}{10} = 1\]Using the standard normal distribution table, \(P(-1 < Z < 1) = P(Z < 1) - P(Z < -1) \approx 0.8413 - 0.1587\). Thus, \(P(60 < X < 80) \approx 0.6826\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Random Variable
In the context of probability and statistics, a random variable serves as a bridge between mathematics and real-world scenarios. It's a numerical value that is determined by the outcome of a random process. Think of flipping a coin or rolling a die. Each possible outcome corresponds to a different number.
- Types: There are two main types of random variables: discrete and continuous. Discrete random variables can take on a countable number of values, while continuous ones can assume an infinite number of values within a given range.
- Example: If we consider the height of students in a classroom, the distribution of these heights will form a continuous random variable.
- Role in Normal Distribution: In our given exercise, the random variable \(X\) follows a normal distribution, characterized by the bell-shaped curve, which means it spreads out in a predictable pattern defined by its mean and standard deviation.
Standard Deviation
Standard deviation is a critical concept in statistics that measures the amount of variation or dispersion in a set of values. It's essentially the average distance of each data point from the mean of the data set.
- Formula: The formula for standard deviation uses the mean of the data set to calculate the variance, and the square root of this variance gives us the standard deviation.
- Significance: A low standard deviation indicates that the values tend to be very close to the mean, while a high standard deviation means the values are spread out over a wider range.
- Application in Exercise: In our original exercise, \(\sigma = 10\), indicates how much individual values of \(X\) deviate from the mean of 70.
Z-score
A Z-score answers a simple yet important question: "How many standard deviations is a value from the mean?" It's a way to transform your data into the standard normal distribution.
- Formula: The formula \(Z = \frac{X - \mu}{\sigma}\) is used, where \(\mu\) is the mean and \(\sigma\) is the standard deviation.
- Understanding: A Z-score measures the number of standard deviations a data point is from the mean. For instance, a Z-score of 2 indicates that the data point is 2 standard deviations above the mean.
- Role in Example: We use Z-scores in the exercise to convert \(X\) values (like 50, 60, 90) into a standard form, which helps in finding probabilities using the standard normal distribution table.
Probability Calculation
Probability calculation involves determining the likelihood of a specific outcome or range of outcomes occurring in a random experiment.
- General Idea: Probability values range between 0 and 1, where 0 means the event will not occur and 1 means it will certainly occur.
- Using Normal Distribution: For a normally distributed random variable, probabilities are determined using the Z-score and standard normal distribution table.
- Application in Exercise: To find different probabilities, like \(P(X>50)\) or \(P(60 < X < 80)\), we first compute the corresponding Z-scores and then find their related probabilities from the standard normal distribution table.
