---## Statistical Problems: Z-Scores and Probability### Problem Set1. **Declare all variables and provide a sketch of the following problems:** **a) Evaluate $ P(z > 0.97) $. Provide a sketch.** - **Explanation:** Determine the probability that a standard normal random variable $ z $ is greater than 0.97. The associated sketch would typically show the standard normal distribution curve with the area to the right of $ z = 0.97 $ shaded to indicate the probability. **b) Find the area between the following z-scores: $ z = 1.49 $ and $ z = 2.3 $. Express the area as a probability statement and provide a sketch.** - **Explanation:** Calculate the probability that a standard normal random variable $ z $ falls between 1.49 and 2.3. The sketch should depict the standard normal distribution curve with the area between $ z = 1.49 $ and $ z = 2.3 $ shaded. **c) Find $ z_{70} $. Assume that $ z_{\alpha} $ represents a z-score with area $ \alpha $ to its right.** - **Explanation:** Identify the z-score such that 70% of the data is to the left (or equivalently, 30% is to the right) of this z-score in the standard normal distribution.### Diagram Explanations- **Standard Normal Distribution Sketches:** Typically represent a bell-shaped curve symmetric around zero. Areas are shaded based on the z-scores provided to represent probabilities related to given questions.---This text is designed to guide students through problems on evaluating and interpreting z-scores within the context of probability, providing visual support through sketch references.

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1. Declare all variables and provide a sketch of the following problems.

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Concept explainers

Addition Rule of Probability

It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.

Expected Value

When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).

Probability Distributions

Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.

Basic Probability

The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.

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Question

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## Statistical Problems: Z-Scores and Probability

### Problem Set

1. **Declare all variables and provide a sketch of the following problems:**

**a) Evaluate $ P(z > 0.97) $. Provide a sketch.**

- **Explanation:** Determine the probability that a standard normal random variable $ z $ is greater than 0.97. The associated sketch would typically show the standard normal distribution curve with the area to the right of $ z = 0.97 $ shaded to indicate the probability.

**b) Find the area between the following z-scores: $ z = 1.49 $ and $ z = 2.3 $. Express the area as a probability statement and provide a sketch.**

- **Explanation:** Calculate the probability that a standard normal random variable $ z $ falls between 1.49 and 2.3. The sketch should depict the standard normal distribution curve with the area between $ z = 1.49 $ and $ z = 2.3 $ shaded.

**c) Find $ z_{70} $. Assume that $ z_{\alpha} $ represents a z-score with area $ \alpha $ to its right.**

- **Explanation:** Identify the z-score such that 70% of the data is to the left (or equivalently, 30% is to the right) of this z-score in the standard normal distribution.

### Diagram Explanations

- **Standard Normal Distribution Sketches:** Typically represent a bell-shaped curve symmetric around zero. Areas are shaded based on the z-scores provided to represent probabilities related to given questions.

---

This text is designed to guide students through problems on evaluating and interpreting z-scores within the context of probability, providing visual support through sketch references.

Expert Solution

Step 1

The sketch of the P(z > 0.97 ) using the statistical tool is,

Thus, P(z> 0.97) =0.1660

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