A population of values has a normal distribution with u = 35.2 and o = 28.7. You intend to draw a random sample of size n = 44. Please show your answers as numbers accurate to 4 decimal places. Find the probability that a single randomly selected value is greater than 33. P(X > 33) = Find the probability that a sample of size n = 44 is randomly selected with a mean greater than 33. P(E > 33) =

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### Understanding Normal Distributions: Sample Problems

A population of values has a normal distribution with a mean (μ) of 35.2 and a standard deviation (σ) of 28.7. You intend to draw a random sample of size \( n = 44 \). Please show your answers as numbers accurate to four decimal places.

#### 1. Probability for a Single Randomly Selected Value
Find the probability that a single randomly selected value is greater than 33.
\[ P(X > 33) = \]

#### 2. Probability for a Sample Mean
Find the probability that a sample of size \( n = 44 \) is randomly selected with a mean greater than 33.
\[ P(\bar{x} > 33) = \]

To solve these problems, one typically utilizes z-scores and standard normal distribution tables (or software). Here is how the problems might be approached:

1. **Calculating \( P(X > 33) \):**
   - Compute the z-score for \( X = 33 \) using the formula: 
     \[ Z = \frac{X - \mu}{\sigma} \]

2. **Calculating \( P(\bar{x} > 33) \):**
   - Determine the standard error of the mean (SEM): 
     \[ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \]
   - Compute the z-score for \( \bar{x} = 33 \) using the formula: 
     \[ Z = \frac{\bar{x} - \mu}{\sigma_{\bar{x}}} \]

### Graphs and Diagrams
Though not provided in the above excerpt, for educational purposes, you may include visual aids such as:

- **Normal Distribution Curve for \( P(X > 33) \)**:
  Illustrate a normal distribution curve where the mean \( \mu = 35.2 \) and show the area to the right of \( X = 33 \).

- **Sampling Distribution of the Sample Mean for \( P(\bar{x} > 33) \)**:
  Depict the distribution of sample means (with reduced spread due to SEM) and highlight the area to the right of \( \bar{x} = 33 \).

These visual representations can help students intuitively understand the probabilities associated with normal distributions and sampling distributions.
Transcribed Image Text:### Understanding Normal Distributions: Sample Problems A population of values has a normal distribution with a mean (μ) of 35.2 and a standard deviation (σ) of 28.7. You intend to draw a random sample of size \( n = 44 \). Please show your answers as numbers accurate to four decimal places. #### 1. Probability for a Single Randomly Selected Value Find the probability that a single randomly selected value is greater than 33. \[ P(X > 33) = \] #### 2. Probability for a Sample Mean Find the probability that a sample of size \( n = 44 \) is randomly selected with a mean greater than 33. \[ P(\bar{x} > 33) = \] To solve these problems, one typically utilizes z-scores and standard normal distribution tables (or software). Here is how the problems might be approached: 1. **Calculating \( P(X > 33) \):** - Compute the z-score for \( X = 33 \) using the formula: \[ Z = \frac{X - \mu}{\sigma} \] 2. **Calculating \( P(\bar{x} > 33) \):** - Determine the standard error of the mean (SEM): \[ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \] - Compute the z-score for \( \bar{x} = 33 \) using the formula: \[ Z = \frac{\bar{x} - \mu}{\sigma_{\bar{x}}} \] ### Graphs and Diagrams Though not provided in the above excerpt, for educational purposes, you may include visual aids such as: - **Normal Distribution Curve for \( P(X > 33) \)**: Illustrate a normal distribution curve where the mean \( \mu = 35.2 \) and show the area to the right of \( X = 33 \). - **Sampling Distribution of the Sample Mean for \( P(\bar{x} > 33) \)**: Depict the distribution of sample means (with reduced spread due to SEM) and highlight the area to the right of \( \bar{x} = 33 \). These visual representations can help students intuitively understand the probabilities associated with normal distributions and sampling distributions.
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