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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Question
### Critical Values and Standard Normal Distribution #### Critical Values for Hypothesis Testing The diagram at the top shows the critical values for various significance levels (\(\alpha\)) in one-tailed and two-tailed tests: 1. **One-tailed (left) Test:** - \(\alpha = 0.05\), critical z = -1.64 - \(\alpha = 0.01\), critical z = -2.33 - \(\alpha = 0.001\), critical z = -3.08 *The shaded area represents the critical region under the curve to the left.* 2. **One-tailed (right) Test:** - \(\alpha = 0.05\), critical z = 1.64 - \(\alpha = 0.01\), critical z = 2.33 - \(\alpha = 0.001\), critical z = 3.08 *The shaded area represents the critical region under the curve to the right.* 3. **Two-tailed Test:** - \(\alpha = 0.05\), critical z = ±1.96 - \(\alpha = 0.01\), critical z = ±2.57 - \(\alpha = 0.001\), critical z = ±3.32 *The shaded areas represent the critical regions under both tails of the curve.* #### Standard Normal Table The table below the diagrams provides values for the cumulative distribution function of the standard normal distribution. This table allows you to find the probability that a standard normal random variable is less than or equal to a given value (Z): - The table is structured in a grid format, with Z-scores ranging from 0.0 to 2.9. - Each cell within the table represents the cumulative probability from the standard normal distribution. For example, at Z = 1.0 and 0.05, the cumulative probability is 0.5199. This information is essential for determining the probability associated with a Z-score in statistical analyses, particularly hypothesis testing and confidence interval estimation. It helps in finding critical values needed to make informed decisions in research studies.
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Transcribed Image Text:### Critical Values and Standard Normal Distribution #### Critical Values for Hypothesis Testing The diagram at the top shows the critical values for various significance levels (\(\alpha\)) in one-tailed and two-tailed tests: 1. **One-tailed (left) Test:** - \(\alpha = 0.05\), critical z = -1.64 - \(\alpha = 0.01\), critical z = -2.33 - \(\alpha = 0.001\), critical z = -3.08 *The shaded area represents the critical region under the curve to the left.* 2. **One-tailed (right) Test:** - \(\alpha = 0.05\), critical z = 1.64 - \(\alpha = 0.01\), critical z = 2.33 - \(\alpha = 0.001\), critical z = 3.08 *The shaded area represents the critical region under the curve to the right.* 3. **Two-tailed Test:** - \(\alpha = 0.05\), critical z = ±1.96 - \(\alpha = 0.01\), critical z = ±2.57 - \(\alpha = 0.001\), critical z = ±3.32 *The shaded areas represent the critical regions under both tails of the curve.* #### Standard Normal Table The table below the diagrams provides values for the cumulative distribution function of the standard normal distribution. This table allows you to find the probability that a standard normal random variable is less than or equal to a given value (Z): - The table is structured in a grid format, with Z-scores ranging from 0.0 to 2.9. - Each cell within the table represents the cumulative probability from the standard normal distribution. For example, at Z = 1.0 and 0.05, the cumulative probability is 0.5199. This information is essential for determining the probability associated with a Z-score in statistical analyses, particularly hypothesis testing and confidence interval estimation. It helps in finding critical values needed to make informed decisions in research studies.
**Question 2: Potency of Antiviral Pills** A population of individual antiviral pills is measured based on their potency. The potency follows a normal distribution with: - Mean: 20 micrograms of active medicine - Standard deviation: 2 micrograms Determine which of the following options represents the range of active medicine that includes 68% of the population of pills: a. 20 to 22 micrograms b. 19 to 21 micrograms c. 18 to 20 micrograms d. 18 to 22 micrograms e. 16 to 24 micrograms **Explanation**: The range that includes 68% of the population in a normal distribution is typically within one standard deviation from the mean.
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Transcribed Image Text:**Question 2: Potency of Antiviral Pills** A population of individual antiviral pills is measured based on their potency. The potency follows a normal distribution with: - Mean: 20 micrograms of active medicine - Standard deviation: 2 micrograms Determine which of the following options represents the range of active medicine that includes 68% of the population of pills: a. 20 to 22 micrograms b. 19 to 21 micrograms c. 18 to 20 micrograms d. 18 to 22 micrograms e. 16 to 24 micrograms **Explanation**: The range that includes 68% of the population in a normal distribution is typically within one standard deviation from the mean.
Expert Solution
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Step 1

Solution-:

Let, X= The potency of individual antiviral pills

Given: \mu=20, \sigma=2

\therefore X \rightarrow N(\mu=20 ,\sigma^2=2^2 )

We find minimum and maximum level of active medicine that includes 68% of the population of pills

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