**Understanding Probabilities and Unusual Sample Means**Welcome to our educational resource. In this section, we will guide you in assessing probabilities and determining the unusualness of sample means in a given data set.### Example Problem#### Given Data:- **Population Mean (μ):** 12,750- **Population Standard Deviation (σ):** 1.8- **Sample Size (n):** 38#### Problem Statement:Find the probability that a sample mean is either less than 12,750 or greater than 12,753 and determine if this range is considered unusual.### Steps to Solve:1. **Calculate the Standard Error of the Mean (SEM):**\[ \text{SEM} = \frac{\sigma}{\sqrt{n}} = \frac{1.8}{\sqrt{38}} \]2. **Determine Z-Scores:** - For $ x = 12,750 $: \[ Z = \frac{12,750 - 12,750}{\text{SEM}} = 0 \] - For $ x = 12,753 $: \[ Z = \frac{12,753 - 12,750}{\text{SEM}} \]3. **Find Probabilities:** - Use the Z-scores to find corresponding probabilities from the standard normal distribution table or using technology tools. - Sum the probabilities for $ x < 12,750 $ and $ x > 12,753 $.4. **Determination of Unusualness:** - Compare the calculated probability with the threshold (0.05). - Based on this comparison, decide if the sample mean falls into the unusual range.### Calculated Probability:\[ \text{Probability} = \]_Round your answer to four decimal places as a precise probability measurement is required._### Decision:Would the given sample mean be considered unusual?**Options:**A. The sample mean would not be considered unusual because there is a probability greater than 0.05 of the sample mean being within this range.B. The sample mean would be considered unusual because there is a probability greater than 0.05 of the sample mean being within this range.C. The sample mean would not be considered unusual because there is a probability less than 0.05 of the sample mean being within this range.D. The sample mean would

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Description: **Understanding Probabilities and Unusual Sample Means**Welcome to our educational resource. In this section, we will guide you in assessing probabilities and determining the unusualness of sample means in a given data set.### Example Problem#### Gi

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d to four decimal places as needed.) the given sample mean be considered unusual? The sample mean would not be considered unusual because there is a probability greater than 0.05 of the sample mean beir The sample mean would be considered unusual because there is a probability greater than 0.05 of the sample mean being w The sample mean would not be considered unusual because there is a probability less than 0.05 of the sample mean being

d to four decimal places as needed.) the given sample mean be considered unusual? The sample mean would not be considered unusual because there is a probability greater than 0.05 of the sample mean beir The sample mean would be considered unusual because there is a probability greater than 0.05 of the sample mean being w The sample mean would not be considered unusual because there is a probability less than 0.05 of the sample mean being

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

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Concept explainers

Inverse Normal Distribution

The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.

Mean, Median, Mode

It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.

Z-Scores

A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.

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Question

**Understanding Probabilities and Unusual Sample Means**

Welcome to our educational resource. In this section, we will guide you in assessing probabilities and determining the unusualness of sample means in a given data set.

### Example Problem

#### Given Data:
- **Population Mean (μ):** 12,750
- **Population Standard Deviation (σ):** 1.8
- **Sample Size (n):** 38

#### Problem Statement:
Find the probability that a sample mean is either less than 12,750 or greater than 12,753 and determine if this range is considered unusual.

### Steps to Solve:

1. **Calculate the Standard Error of the Mean (SEM):**

\[ \text{SEM} = \frac{\sigma}{\sqrt{n}} = \frac{1.8}{\sqrt{38}} \]

2. **Determine Z-Scores:**

- For $ x = 12,750 $:

\[ Z = \frac{12,750 - 12,750}{\text{SEM}} = 0 \]

- For $ x = 12,753 $:

\[ Z = \frac{12,753 - 12,750}{\text{SEM}} \]

3. **Find Probabilities:**
- Use the Z-scores to find corresponding probabilities from the standard normal distribution table or using technology tools.
- Sum the probabilities for $ x < 12,750 $ and $ x > 12,753 $.

4. **Determination of Unusualness:**
- Compare the calculated probability with the threshold (0.05).
- Based on this comparison, decide if the sample mean falls into the unusual range.

### Calculated Probability:
\[ \text{Probability} = \]

_Round your answer to four decimal places as a precise probability measurement is required._

### Decision:

Would the given sample mean be considered unusual?

**Options:**

A. The sample mean would not be considered unusual because there is a probability greater than 0.05 of the sample mean being within this range.

B. The sample mean would be considered unusual because there is a probability greater than 0.05 of the sample mean being within this range.

C. The sample mean would not be considered unusual because there is a probability less than 0.05 of the sample mean being within this range.

D. The sample mean would

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