A population of values has a normal distribution with u = 95.4 and o = 45.4. A random sample of size n = 95 is drawn. a. Find the probability that a single randomly selected value is between 81 and 81.4. Round your answer to four decimal places. P(81 < X < 81.4) = b. Find the probability that a sample of size n = 95 is randomly selected with a mean between 81 and 81.4. Round your answer to four decimal places. P(81 < M < 81.4) : %3D

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## Probability and Statistics: Normal Distribution Problem

A population of values has a normal distribution with a mean (\(\mu\)) of 95.4 and a standard deviation (\(\sigma\)) of 45.4. A random sample of size \(n = 95\) is drawn.

### Problem Statement

#### a. Probability for a Single Randomly Selected Value

Find the probability that a single randomly selected value \(X\) lies between 81 and 81.4. Round your answer to four decimal places.

\[
P(81 < X < 81.4) = \, \underline{\phantom{answer}} 
\]

#### b. Probability for a Sample Mean

Find the probability that a sample mean \(M\) for \(n = 95\) is randomly selected with a mean between 81 and 81.4. Round your answer to four decimal places.

\[
P(81 < M < 81.4) = \, \underline{\phantom{answer}}
\]

### Additional Information

- **Submit Question**: Once you have calculated the probabilities, you can submit your answers using the "Submit Question" button provided.

This exercise will help you understand how to use normal distributions to calculate probabilities for individual values and sample means within specified ranges. Make sure to use the properties of the normal distribution and possibly standard normal tables or calculators to find the required probabilities.
Transcribed Image Text:## Probability and Statistics: Normal Distribution Problem A population of values has a normal distribution with a mean (\(\mu\)) of 95.4 and a standard deviation (\(\sigma\)) of 45.4. A random sample of size \(n = 95\) is drawn. ### Problem Statement #### a. Probability for a Single Randomly Selected Value Find the probability that a single randomly selected value \(X\) lies between 81 and 81.4. Round your answer to four decimal places. \[ P(81 < X < 81.4) = \, \underline{\phantom{answer}} \] #### b. Probability for a Sample Mean Find the probability that a sample mean \(M\) for \(n = 95\) is randomly selected with a mean between 81 and 81.4. Round your answer to four decimal places. \[ P(81 < M < 81.4) = \, \underline{\phantom{answer}} \] ### Additional Information - **Submit Question**: Once you have calculated the probabilities, you can submit your answers using the "Submit Question" button provided. This exercise will help you understand how to use normal distributions to calculate probabilities for individual values and sample means within specified ranges. Make sure to use the properties of the normal distribution and possibly standard normal tables or calculators to find the required probabilities.
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