Given X ~ N(μ, σ²) with P(X < 62.8) = 0.90 and P(X < 66.5) = 0.95, how do you find the mean μ?

To find μ, identify the z-scores corresponding to 0.90 and 0.95, then set up equations using the inverse normal function: μ = x - zσ. Using the z-scores for 0.90 (~1.28) and 0.95 (~1.64), solve the system to find μ.

What is the significance of the probabilities P(X < 62.8) = 0.90 and P(X < 66.5) = 0.95 in this problem?

They specify the cumulative distribution function (CDF) values at specific points, allowing us to find the corresponding z-scores and subsequently the mean and variance.

How do you determine the z-scores associated with the given probabilities?

Use the standard normal table or inverse normal function: for 0.90, z ≈ 1.28; for 0.95, z ≈ 1.64.

Once the z-scores are known, how do you compute the mean μ?

Use the formula x = μ + zσ. Rearranged, μ = x - zσ. Plug in the x-values and the z-scores to solve for μ once σ is known or expressed in terms of μ.

How do you find the variance σ² in this problem?

Use the two points and their corresponding z-scores to set up two equations. Solving these equations simultaneously yields σ, and squaring it gives σ².

Can you explain the step-by-step process to find μ and σ² from the given probabilities?

Yes. First, identify z-scores for the given probabilities (1.28 and 1.64). Then, set up equations: 62.8 = μ + 1.28σ and 66.5 = μ + 1.64σ. Solve these simultaneously to find μ and σ. Finally, compute σ².

What assumptions are made when solving for μ and σ² in this problem?

The main assumption is that X follows a normal distribution N(μ, σ²), which allows us to use z-scores and properties of the normal distribution to find parameters.

Is there a shortcut to find μ and σ² without solving complex equations directly?

Typically, you need to set up and solve the equations using the z-scores and the given x-values. No significant shortcut exists beyond using the standard normal table and algebraic manipulation.

What is the final step after calculating μ and σ² for this problem?

Verify the results by checking if the calculated μ and σ² satisfy the original probability statements, ensuring consistency.

Let X Be N (u, A^2) So That P(X &lt; 62.8) = 0.90 And P(X &lt; 66.5) = 0.95. Find U And A^2

Q: Given X ~ N(μ, σ²) with P(X < 62.8) = 0.90 and P(X < 66.5) = 0.95, how do you find the mean μ?

To find μ, identify the z-scores corresponding to 0.90 and 0.95, then set up equations using the inverse normal function: μ = x - zσ. Using the z-scores for 0.90 (~1.28) and 0.95 (~1.64), solve the system to find μ.

Q: What is the significance of the probabilities P(X < 62.8) = 0.90 and P(X < 66.5) = 0.95 in this problem?

They specify the cumulative distribution function (CDF) values at specific points, allowing us to find the corresponding z-scores and subsequently the mean and variance.

Q: How do you determine the z-scores associated with the given probabilities?

Use the standard normal table or inverse normal function: for 0.90, z ≈ 1.28; for 0.95, z ≈ 1.64.

Q: Once the z-scores are known, how do you compute the mean μ?

Use the formula x = μ + zσ. Rearranged, μ = x - zσ. Plug in the x-values and the z-scores to solve for μ once σ is known or expressed in terms of μ.

Q: How do you find the variance σ² in this problem?

Use the two points and their corresponding z-scores to set up two equations. Solving these equations simultaneously yields σ, and squaring it gives σ².

Q: Can you explain the step-by-step process to find μ and σ² from the given probabilities?

Yes. First, identify z-scores for the given probabilities (1.28 and 1.64). Then, set up equations: 62.8 = μ + 1.28σ and 66.5 = μ + 1.64σ. Solve these simultaneously to find μ and σ. Finally, compute σ².

Q: What assumptions are made when solving for μ and σ² in this problem?

The main assumption is that X follows a normal distribution N(μ, σ²), which allows us to use z-scores and properties of the normal distribution to find parameters.

Q: Is there a shortcut to find μ and σ² without solving complex equations directly?

Typically, you need to set up and solve the equations using the z-scores and the given x-values. No significant shortcut exists beyond using the standard normal table and algebraic manipulation.

Q: What is the final step after calculating μ and σ² for this problem?

Verify the results by checking if the calculated μ and σ² satisfy the original probability statements, ensuring consistency.

Let X Be N (u, A^2) So That P(X < 62.8) = 0.90 And P(X < 66.5) = 0.95. Find U And A^2

Understanding the properties of the normal distribution is fundamental in statistics, especially when analyzing data and making inferences about population parameters. In this article, we will explore how to determine the mean (μ) and variance (σ²) of a normal distribution given specific probability statements. We will systematically approach this problem, explaining the underlying concepts and demonstrating the step-by-step calculations involved.

Introduction to Normal Distribution

The normal distribution, often called the bell curve, is a continuous probability distribution characterized by its symmetric shape centered around the mean. It is defined by two parameters:

The mean (μ), which indicates the center of the distribution.

The variance (σ²), which measures the spread or dispersion of the data.

Mathematically, the probability density function (pdf) of a normal distribution is expressed as:

\[
f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left( -\frac{(x - \mu)^2}{2\sigma^2} \right)
\]

Understanding how to work with this distribution involves familiarity with standardization and the standard normal distribution.

Standard Normal Distribution and Z-Score

The standard normal distribution, denoted as Z ~ N(0,1), is a special case of the normal distribution with a mean of 0 and variance of 1. To relate any normal distribution N(μ, σ²) to the standard normal, we use the Z-score transformation:

\[
Z = \frac{X - \mu}{\sigma}
\]

This transformation allows us to find probabilities associated with X by referring to the standard normal distribution.

Given Probabilities and Their Significance

In our problem, we are given two probability statements:

\( P(X < 62.8) = 0.90 \)

\( P(X < 66.5) = 0.95 \)

These statements tell us that:

The 90th percentile of the distribution is 62.8.

The 95th percentile of the distribution is 66.5.

Our goal is to determine the mean (μ) and variance (σ²) based on these percentiles.

Step-by-Step Solution Approach

To find μ and σ², follow these steps:

Convert the given probabilities into Z-scores using standard normal distribution tables or inverse functions.

Set up equations linking the Z-scores to the unknown μ and σ.

Solve the system of equations to find μ and σ.

Calculate the variance as σ².

Let's delve into each step in detail.

Step 1: Find Corresponding Z-Scores

The probabilities correspond to specific Z-scores in the standard normal distribution:

For \( P(Z < z_1) = 0.90 \)

For \( P(Z < z_2) = 0.95 \)

Using standard normal distribution tables or statistical software:

\( z_{0.90} \approx 1.2816 \)

\( z_{0.95} \approx 1.6449 \)

These Z-scores indicate the number of standard deviations the data points are from the mean in the standard normal distribution.

Step 2: Relate Z-Scores to X Values

Using the Z-score formula:

\[
Z = \frac{X - \mu}{\sigma}
\]

we can set up two equations:

\[
\frac{62.8 - \mu}{\sigma} = 1.2816
\]
\[
\frac{66.5 - \mu}{\sigma} = 1.6449
\]

These equations relate the unknown parameters μ and σ to known quantities.

Step 3: Solve the Equations for μ and σ

Subtract the first equation from the second:

\[
\frac{66.5 - \mu}{\sigma} - \frac{62.8 - \mu}{\sigma} = 1.6449 - 1.2816
\]

Simplify:

\[
\frac{(66.5 - \mu) - (62.8 - \mu)}{\sigma} = 0.3633
\]

\[
\frac{66.5 - \mu - 62.8 + \mu}{\sigma} = 0.3633
\]

\[
\frac{3.7}{\sigma} = 0.3633
\]

Solve for σ:

\[
\sigma = \frac{3.7}{0.3633} \approx 10.18
\]

Now, substitute σ back into one of the earlier equations to find μ:

\[
\frac{62.8 - \mu}{10.18} = 1.2816
\]

Multiply both sides by 10.18:

\[
62.8 - \mu = 1.2816 \times 10.18 \approx 13.05
\]

Solve for μ:

\[
\mu = 62.8 - 13.05 \approx 49.75
\]

Step 4: Calculate Variance

Since the variance is the square of the standard deviation:

\[
\sigma^2 = (10.18)^2 \approx 103.63
\]

Final Results:

The mean (μ) is approximately 49.75.

The variance (σ²) is approximately 103.63.

Summary of Findings

| Parameter | Estimated Value |
| --- | --- |
| Mean (μ) | 49.75 |
| Variance (σ²) | 103.63 |

These parameters characterize the normal distribution that satisfies the given probability conditions.

Conclusion

In this comprehensive analysis, we demonstrated how to determine the mean and variance of a normal distribution based on specific percentile probabilities. The key steps involved converting probabilities into Z-scores, relating these Z-scores to the original data points, and solving the resulting equations. Mastery of these techniques is essential for statisticians and data analysts working with normally distributed data, enabling them to infer distribution parameters precisely and efficiently.

Understanding this process enhances your ability to interpret statistical data, perform hypothesis testing, and construct confidence intervals — all fundamental tools in the arsenal of modern statistics.