Table of contents

1. Intro to Stats and Collecting Data55m
2. Describing Data with Tables and Graphs1h 55m
3. Describing Data Numerically1h 45m
4. Probability2h 16m
5. Binomial Distribution & Discrete Random Variables2h 33m
6. Normal Distribution and Continuous Random Variables1h 38m
7. Sampling Distributions & Confidence Intervals: Mean1h 53m
8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- Sampling Distribution of Sample Proportion
  29m
- Confidence Intervals for Population Proportion
  42m
9. Hypothesis Testing for One Sample2h 19m
10. Hypothesis Testing for Two Samples3h 22m
11. Correlation1h 6m
12. Regression1h 4m
13. Chi-Square Tests & Goodness of Fit1h 20m
14. ANOVA1h 0m

6. Normal Distribution and Continuous Random Variables

Standard Normal Distribution

Statistics

6. Normal Distribution and Continuous Random Variables

Standard Normal Distribution

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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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Related Practice

Textbook Question

In Exercises 21–24, a control chart is shown. Each chart has horizontal lines drawn at the mean mu, at mu ±2sigma, and at mu±3sigma. Determine whether the process shown is in control or out of control. Explain.

An engine part has been designed to have a diameter of 55 millimeters. The standard deviation of the process is 0.001 millimeter.

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Textbook Question

Construction About 63% of the residents in a town are in favor of building a new high school. One hundred five residents are randomly selected. What is the probability that the sample proportion in favor of building a new school is less than 55%? Interpret your result.

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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.

P= 0.2802

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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

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Textbook Question

"According to data from the city of Toronto, Ontario, Canada, there were nearly 112,000 parking infractions in the city for December 2020, with fines totaling over 5,500,000 Canadian dollars. The fines (in Canadian dollars) for a random sample of 105 parking infractions in Toronto, Ontario, Canada, for December 2020 are listed below. (Source: City of Toronto)

In Exercises 1–5, use technology. If possible, print your results.

Draw a histogram for the data. Does the distribution appear to be bell-shaped?"

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Multiple Choice

Find the area under the standard normal distribution to the left of a z-score of $1.21$ .

Find the area under the standard normal distribution to the right of a z-score of $-0.44$ .

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