Which Interval Has The Greatest Probability In A Standard Normal Distribution?

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Determining which probability range is the greatest for a standard normal distribution requires a clear understanding of the properties of the standard normal curve. This article delves into the question, "Which of the following probabilities is the greatest for a standard normal distribution?" analyzing the given options and providing a comprehensive explanation. We will explore the symmetry of the standard normal distribution, the concept of area under the curve, and how these principles help us identify the interval with the highest probability.

Understanding the Standard Normal Distribution

To address the question of greatest probability in a standard normal distribution, it is crucial to first grasp the fundamental characteristics of this distribution. The standard normal distribution is a specific type of normal distribution characterized by a mean of 0 and a standard deviation of 1. Its graphical representation, the standard normal curve, is a symmetrical bell-shaped curve centered around the mean. This symmetry is a key property that influences probabilities associated with different intervals along the z-axis.

Key Properties of the Standard Normal Distribution

  1. Symmetry: The curve is perfectly symmetrical around the mean (z = 0). This implies that the probability of a value falling within a certain distance to the left of the mean is the same as the probability of a value falling within the same distance to the right of the mean. Mathematically, this can be expressed as P(z ≤ -a) = P(z ≥ a) for any value 'a'. Understanding this symmetry is vital when comparing probabilities across different intervals, especially those that are mirror images of each other with respect to the mean.

  2. Total Area: The total area under the standard normal curve is equal to 1. This represents the total probability of all possible outcomes. When we calculate probabilities for specific intervals, we are essentially finding the area under the curve within those intervals. The larger the area, the higher the probability. This concept is fundamental to comparing probabilities of different intervals.

  3. Area and Probability: The area under the curve between any two points on the z-axis represents the probability of a randomly selected value falling within that range. For example, the area under the curve between z = a and z = b represents P(a ≤ z ≤ b). To solve problems involving probabilities, we often refer to standard normal distribution tables (z-tables) or statistical software, which provide these area values for various z-scores. The ability to interpret and apply these values is essential for accurately determining probabilities.

  4. Z-Scores: The z-score represents the number of standard deviations a particular value is away from the mean. A z-score of 1 indicates a value one standard deviation above the mean, while a z-score of -1 indicates a value one standard deviation below the mean. The further a z-score is from 0 (in either direction), the less likely the corresponding value is to occur. This is because the area under the curve decreases as we move away from the center.

Implications for Probability

Given the symmetrical nature of the standard normal distribution and the concentration of data around the mean, intervals closer to the mean will generally have higher probabilities than intervals farther away. This is because the bell-shaped curve is highest at the center and tapers off as we move towards the tails. Understanding this principle is crucial when comparing intervals and determining which one contains the greatest probability. Specifically, intervals of the same width will have different probabilities depending on their location relative to the mean. Intervals centered around the mean will have the highest probabilities, while intervals located in the tails will have lower probabilities.

In summary, a solid understanding of the standard normal distribution's properties, particularly its symmetry and the relationship between area and probability, is essential for solving problems that involve comparing probabilities across different intervals. By keeping these concepts in mind, we can approach the given question with a clear strategy and arrive at the correct answer.

Analyzing the Given Options

When tackling the question of identifying the greatest probability range in a standard normal distribution, a systematic analysis of each provided option is essential. The options present different intervals along the z-axis, and our goal is to determine which interval encompasses the largest area under the standard normal curve. This area directly corresponds to the probability of a z-score falling within that interval. To make an informed comparison, let's examine each option individually and consider its position relative to the mean (z = 0).

The options are:

A. P(-1.5 ≤ z ≤ -0.5) B. P(-0.5 ≤ z ≤ 0.5) C. P(0.5 ≤ z ≤ 1.5) D. P(1.5 ≤ z ≤ 2.5)

Option A: P(-1.5 ≤ z ≤ -0.5)

This interval is located on the left side of the standard normal curve, specifically between 1.5 standard deviations and 0.5 standard deviations below the mean. To visualize this, imagine the area under the curve shaded between these two z-scores. This area represents the probability of a z-score falling within this range. Since this interval is not centered around the mean, and it is relatively far from the mean, we can anticipate that the probability associated with this interval might not be the highest compared to intervals closer to the center.

Option B: P(-0.5 ≤ z ≤ 0.5)

This interval is centered around the mean (z = 0), extending 0.5 standard deviations in both directions. This is a crucial observation. Intervals centered around the mean tend to have higher probabilities because the standard normal curve is at its peak at z = 0. Therefore, the area under the curve within this interval will be substantial. We can reasonably expect that this option might have a higher probability than option A, given its central location.

Option C: P(0.5 ≤ z ≤ 1.5)

This interval is located on the right side of the mean, between 0.5 and 1.5 standard deviations above the mean. Similar to option A, this interval is not centered around the mean. However, it's important to note that the width of this interval (1.5 - 0.5 = 1) is the same as the width of options A and B. The key difference is its location relative to the mean. Since this interval is farther from the mean than the interval in option B, we can expect its probability to be lower.

Option D: P(1.5 ≤ z ≤ 2.5)

This interval is the farthest from the mean among all the options, located between 1.5 and 2.5 standard deviations above the mean. As we move farther away from the mean in either direction, the height of the standard normal curve decreases, resulting in smaller areas under the curve. Therefore, the probability associated with this interval is likely to be the lowest among the given options.

Comparative Analysis

By analyzing each option's location relative to the mean, we can draw some preliminary conclusions. Option B, being centered around the mean, is likely to have the highest probability. Option D, being the farthest from the mean, is likely to have the lowest probability. Options A and C fall somewhere in between, with their probabilities depending on their specific distances from the mean and their widths. To confirm our intuition and arrive at a definitive answer, we need to consider the symmetry of the standard normal distribution and how it affects probabilities in different intervals.

Applying the Properties of Symmetry and Area

To definitively determine the greatest probability among the given options, we must leverage the inherent properties of the standard normal distribution. Specifically, the symmetry of the distribution and the relationship between area under the curve and probability are crucial. Understanding these concepts allows us to compare intervals effectively and accurately identify the one with the highest likelihood.

Symmetry and Probability

The standard normal distribution is perfectly symmetrical around its mean (z = 0). This symmetry implies that the probability of finding a z-score within a certain range to the left of the mean is equal to the probability of finding a z-score within the corresponding range to the right of the mean. Mathematically, this means P(-a ≤ z ≤ 0) = P(0 ≤ z ≤ a) for any value 'a'. This property is invaluable when comparing probabilities of intervals that are mirror images of each other across the mean.

In our case, consider options A and C:

A. P(-1.5 ≤ z ≤ -0.5) C. P(0.5 ≤ z ≤ 1.5)

These intervals are symmetrical about the mean. The interval in option A is a reflection of the interval in option C across the y-axis (z = 0). Therefore, based on the symmetry property, we can conclude that P(-1.5 ≤ z ≤ -0.5) = P(0.5 ≤ z ≤ 1.5). This means that the probabilities associated with options A and C are equal. This eliminates the need to individually calculate these probabilities; we know they are the same.

Area Under the Curve and Probability

The area under the standard normal curve represents probability. The total area under the curve is 1, which represents the total probability of all possible outcomes. The area under the curve between any two z-scores, z1 and z2, represents the probability of a z-score falling within that range, i.e., P(z1 ≤ z ≤ z2). The larger the area under the curve within an interval, the higher the probability.

Given the bell-shaped nature of the standard normal curve, the area is most concentrated around the mean. As we move away from the mean in either direction, the height of the curve decreases, resulting in smaller areas for intervals of the same width. This is a key principle when comparing probabilities of intervals at different locations along the z-axis.

Now, let's compare options B and C:

B. P(-0.5 ≤ z ≤ 0.5) C. P(0.5 ≤ z ≤ 1.5)

Both intervals have the same width (1 unit). However, the interval in option B is centered around the mean, while the interval in option C is located farther away from the mean. Since the area under the curve is greatest near the mean, the probability associated with option B will be higher than the probability associated with option C. This is because the interval in option B captures a region where the curve is higher, resulting in a larger area.

Conclusion from Symmetry and Area Analysis

By applying the properties of symmetry and area, we have established the following:

  1. P(-1.5 ≤ z ≤ -0.5) = P(0.5 ≤ z ≤ 1.5) (due to symmetry)
  2. P(-0.5 ≤ z ≤ 0.5) > P(0.5 ≤ z ≤ 1.5) (due to the concentration of area around the mean)

Therefore, we can conclude that option B, P(-0.5 ≤ z ≤ 0.5), has the greatest probability among the given options. This is because it is centered around the mean, where the area under the standard normal curve is the highest.

Calculating Probabilities (Optional)

While we have already determined the correct answer using the properties of the standard normal distribution, we can further solidify our understanding by calculating the probabilities for each interval. This involves using a standard normal distribution table (z-table) or statistical software to find the areas under the curve corresponding to each option. Although not strictly necessary for answering the question, this step provides a numerical confirmation of our conclusions.

Using a Z-Table

A z-table provides the cumulative probability P(z ≤ a) for various values of 'a'. To find the probability of an interval P(z1 ≤ z ≤ z2), we calculate P(z ≤ z2) - P(z ≤ z1). Let's apply this to each option:

A. P(-1.5 ≤ z ≤ -0.5) = P(z ≤ -0.5) - P(z ≤ -1.5) B. P(-0.5 ≤ z ≤ 0.5) = P(z ≤ 0.5) - P(z ≤ -0.5) C. P(0.5 ≤ z ≤ 1.5) = P(z ≤ 1.5) - P(z ≤ 0.5) D. P(1.5 ≤ z ≤ 2.5) = P(z ≤ 2.5) - P(z ≤ 1.5)

Consulting a z-table, we find the following approximate values:

  • P(z ≤ -1.5) ≈ 0.0668
  • P(z ≤ -0.5) ≈ 0.3085
  • P(z ≤ 0.5) ≈ 0.6915
  • P(z ≤ 1.5) ≈ 0.9332
  • P(z ≤ 2.5) ≈ 0.9938

Now, we can calculate the probabilities for each option:

A. P(-1.5 ≤ z ≤ -0.5) ≈ 0.3085 - 0.0668 ≈ 0.2417 B. P(-0.5 ≤ z ≤ 0.5) ≈ 0.6915 - 0.3085 ≈ 0.3830 C. P(0.5 ≤ z ≤ 1.5) ≈ 0.9332 - 0.6915 ≈ 0.2417 D. P(1.5 ≤ z ≤ 2.5) ≈ 0.9938 - 0.9332 ≈ 0.0606

Numerical Confirmation

The calculated probabilities confirm our earlier analysis based on the properties of symmetry and area. We see that:

  • Option B has the highest probability (approximately 0.3830).
  • Options A and C have equal probabilities (approximately 0.2417), as expected due to symmetry.
  • Option D has the lowest probability (approximately 0.0606), as it is farthest from the mean.

This numerical confirmation reinforces the importance of understanding the underlying principles of the standard normal distribution. While calculating probabilities using a z-table or software is a valuable skill, a strong conceptual understanding allows us to make informed comparisons and predictions without relying solely on numerical calculations.

Conclusion

In conclusion, the greatest probability for a standard normal distribution among the given options is P(-0.5 ≤ z ≤ 0.5). This determination is based on the fundamental properties of the standard normal distribution, particularly its symmetry around the mean and the concentration of area under the curve near the mean. Intervals centered around the mean, like the one in option B, inherently encompass larger areas under the curve, leading to higher probabilities.

By analyzing the options and applying these principles, we effectively compared the intervals and identified the one with the highest probability. Furthermore, calculating the probabilities using a z-table provided a numerical validation of our conclusions, reinforcing the importance of both conceptual understanding and practical calculation skills in working with standard normal distributions.

This exploration highlights the significance of grasping the characteristics of statistical distributions to make informed decisions and solve probability-related problems effectively.