Graphical Analysis In Exercises 9–12, match the P-value or z-stat...

6. Normal Distribution and Continuous Random Variables

Standard Normal Distribution

Statistics

6. Normal Distribution and Continuous Random Variables

Standard Normal Distribution

1:32 minutes

Problem 7.2.11

Textbook Question

Graphical Analysis In Exercises 9–12, match the P-value or z-statistic with the graph that represents the corresponding area. Explain your reasoning.

z = -2.37

Verified step by step guidance

Step 1: Understand the z-statistic value provided in the problem. A z-statistic of -2.37 indicates that the value is 2.37 standard deviations below the mean in a standard normal distribution.

Step 2: Recall that the standard normal distribution is symmetric and centered at zero. Negative z-values correspond to the left side of the distribution.

Step 3: Identify the area under the curve to the left of z = -2.37. This area represents the cumulative probability (P-value) for z = -2.37.

Step 4: Match the graph provided with the z-statistic. The shaded region on the graph corresponds to the cumulative probability for z = -2.37, which is the area to the left of this z-value.

Step 5: To calculate the P-value, you would use a z-table or statistical software to find the cumulative probability associated with z = -2.37. This value represents the proportion of the distribution that lies to the left of z = -2.37.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Z-Score

A z-score indicates how many standard deviations an element is from the mean of a distribution. In this context, a z-score of -2.37 suggests that the value is 2.37 standard deviations below the mean, which is crucial for determining the corresponding area under the normal curve.

P-Value

The p-value represents the probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true. It is used to determine the significance of results in hypothesis testing, with lower p-values indicating stronger evidence against the null hypothesis.

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its bell-shaped curve, defined by its mean and standard deviation. Understanding this distribution is essential for interpreting z-scores and p-values, as it provides the framework for calculating probabilities and areas under the curve.

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Graphical Analysis In Exercises 9–12, match the P-value or z-statistic with the graph that represents the corresponding area. Explain your reasoning.z = -2.37