MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN: 9781119256830
Author: Amos Gilat
Publisher: John Wiley & Sons Inc
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**Statistics: Assignment 17.30**

**Dice Roll Analysis**

**Background:**
In Assignment 11.15, we evaluated the mean and standard deviation for \( X \), which represents the number of dots facing up when a single die is rolled. Here, the mean calculated was \( \mu = 3.5 \) and the standard deviation was \( \sigma = 1.7078 \).

**Scenario:**
Now, consider tossing two dice and calculating the mean number of dots facing up. Let \( \overline{X} \) be the mean number of dots facing up on the two dice. For two dice, \( \overline{X} \) could range from 1.0 to 6.0 in increments of 0.5 (e.g., 1.0, 1.5, ..., 6.0).

**Random Variables:**
The expected value, or mean, of this random variable \( \overline{X} \) is denoted by \( E(\overline{X}) \) or \( \mu_{\overline{X}} \).

**Standard Deviation:**
The standard deviation of \( \overline{X} \), also known as the standard error of the mean, is denoted by \( \sigma_{\overline{X}} \). This is calculated easily if you know \( \sigma \).

**Exercises:**
Complete the sentence:
- When tossing two dice and finding the mean, \( \overline{X} \), the mean or expected value \( \mu_{\overline{X}} = \) ________ and a standard deviation \( \sigma_{\overline{X}} = \) ________.

Next, consider tossing five dice and finding the mean, and how it affects the expected value and standard deviation. Repeat this for samples of 10, 20, and 50 dice.

**Table Completion:**

| Number of dice tossed from which we’ll find the sample mean, \( \overline{X} \) \( (n) \) | Mean or expected value of \( \overline{X} \) \( E(\overline{X}) \) or \( \mu_{\overline{X}} \) | Standard deviation of \( \overline{X} \) \( \sigma_{\overline{X}} \) |
|---|---|---|
| 1 | 3.5
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Transcribed Image Text:**Statistics: Assignment 17.30** **Dice Roll Analysis** **Background:** In Assignment 11.15, we evaluated the mean and standard deviation for \( X \), which represents the number of dots facing up when a single die is rolled. Here, the mean calculated was \( \mu = 3.5 \) and the standard deviation was \( \sigma = 1.7078 \). **Scenario:** Now, consider tossing two dice and calculating the mean number of dots facing up. Let \( \overline{X} \) be the mean number of dots facing up on the two dice. For two dice, \( \overline{X} \) could range from 1.0 to 6.0 in increments of 0.5 (e.g., 1.0, 1.5, ..., 6.0). **Random Variables:** The expected value, or mean, of this random variable \( \overline{X} \) is denoted by \( E(\overline{X}) \) or \( \mu_{\overline{X}} \). **Standard Deviation:** The standard deviation of \( \overline{X} \), also known as the standard error of the mean, is denoted by \( \sigma_{\overline{X}} \). This is calculated easily if you know \( \sigma \). **Exercises:** Complete the sentence: - When tossing two dice and finding the mean, \( \overline{X} \), the mean or expected value \( \mu_{\overline{X}} = \) ________ and a standard deviation \( \sigma_{\overline{X}} = \) ________. Next, consider tossing five dice and finding the mean, and how it affects the expected value and standard deviation. Repeat this for samples of 10, 20, and 50 dice. **Table Completion:** | Number of dice tossed from which we’ll find the sample mean, \( \overline{X} \) \( (n) \) | Mean or expected value of \( \overline{X} \) \( E(\overline{X}) \) or \( \mu_{\overline{X}} \) | Standard deviation of \( \overline{X} \) \( \sigma_{\overline{X}} \) | |---|---|---| | 1 | 3.5
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