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

1:30 minutes

Problem 7.2.12

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 = -0.51

Verified step by step guidance

Step 1: Understand the z-statistic. A z-statistic of -0.51 indicates that the value is 0.51 standard deviations below the mean in a standard normal distribution.

Step 2: Recall that the standard normal distribution is symmetric around the mean (z = 0). Negative z-values correspond to the left side of the distribution.

Step 3: Identify the shaded region in the graph. The blue area represents the cumulative probability to the left of z = -0.51.

Step 4: To find the P-value associated with z = -0.51, you would calculate the cumulative probability using a z-table or statistical software. This gives the proportion of the distribution that lies to the left of z = -0.51.

Step 5: Match the graph with the z-statistic. Since the graph shows the cumulative area to the left of z = -0.51, it correctly represents the P-value for this z-statistic.

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Key Concepts

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

Z-Statistic

The z-statistic is a measure that describes how many standard deviations a data point is from the mean of a distribution. It is calculated by subtracting the mean from the data point and then dividing by the standard deviation. In hypothesis testing, the z-statistic helps determine the likelihood of observing a sample statistic under the null hypothesis.

P-Value

The p-value is the probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true. It quantifies the evidence against the null hypothesis; a smaller p-value indicates stronger evidence. In the context of the z-statistic, the p-value can be derived from the area under the normal distribution curve corresponding to the z-value.

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its bell-shaped curve, defined by its mean and standard deviation. It is symmetric around the mean, with about 68% of the data falling within one standard deviation. Understanding the properties of the normal distribution is crucial for interpreting z-statistics and p-values, as many statistical tests assume normality.

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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 = -0.51