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Chapter 5: Problem 8

Find the indicated area under the standard normal curve. If convenient, use technology to find the area. To the left of \(z=-1.95\)

Short Answer

Expert verified
The area to the left of z = -1.95 is approximately 0.0256.

Step by step solution

01

Understanding the Standard Normal Curve

The standard normal curve is a symmetric, bell-shaped graph with a mean of 0 and a standard deviation of 1. The area under this curve represents probabilities and is equal to 1 (100%). To find the area to the left of a particular z-score, we'll use the standard normal distribution table or technology.
02

Using the Z-Table

A z-score table (standard normal table) provides the area to the left of a given z-score. Locate the z-score of -1.95 in the table by finding the row for -1.9 and the column for 0.05. The intersection gives the area or probability.
03

Reading the Z-Table

According to the standard normal distribution table, the area to the left of a z-score of -1.95 is approximately 0.0256. This means there is a 2.56% probability of a random variable falling to the left of z = -1.95 under the standard normal curve.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Z-Score
A Z-score is a numerical measurement that describes a value's relationship to the mean of a group of values, expressed in terms of standard deviations from the mean. Understanding Z-scores is crucial when dealing with the standard normal distribution, as they help us determine how far away a specific data point is from the average.
  • If the Z-score is 0, it means the data point's value is exactly in line with the mean.
  • A positive Z-score indicates that the data point is above the mean.
  • Conversely, a negative Z-score, like -1.95, denotes a value below the mean.
To find the area to the left of a specific Z-score, tools like the Z-table, which provides the cumulative probability for Z-scores, or software can be used. This approach is pivotal when calculating probabilities related to normal distributions.
Probability
Probability is a fundamental concept in statistics that measures the likelihood of an event occurring. In the context of a standard normal distribution, probability is represented as the area under the curve. This entire area under the curve sums up to 1, which translates to a 100% chance of the data falling somewhere on the curve. Finding the probability to the left of a Z-score involves:
  • Identifying the Z-score, which indicates how many standard deviations away a point is from the mean.
  • Using tools like the Z-table to find the cumulative probability up to that Z-score.
In our exercise, the probability of a data point falling to the left of Z = -1.95 is approximately 0.0256, or 2.56%. This tells us there's a low chance of randomly selecting a value less than this Z-score, given the normal distribution.
Normal Curve
The normal curve, often referred to as the bell curve, is a visual representation of the normal distribution. It is symmetric about the mean, depicting that most data points cluster around the center with fewer points occurring as you move away from the mean.
Some key characteristics of the standard normal curve include:
  • A mean of 0, implying the peak of the curve is at this point.
  • A standard deviation of 1, showing how spread out the values are from the mean.
  • The total area under the curve equals 1, reflecting the total probability.
The normal curve is essential in statistics as it allows for the determination of probabilities and the assessment of data's overall distribution. In practice, when confronting any probability-related question, understanding the area relative to the Z-score on this curve can provide insights into data tendencies.
Statistics Education
Statistics education empowers individuals to make informed decisions by understanding and interpreting data effectively. Learning about topics like Z-scores and probability not only helps students excel in their studies but also equips them to analyze real-world scenarios and predict outcomes.
  • Critical thinking skills: Statistics teach students to question data sources, calculations, and conclusions.
  • Data literacy: In today's data-driven world, being able to summarize and interpret data accurately is invaluable.
  • Application of concepts like the normal curve help in diverse fields, from business and economics to healthcare and engineering.
    • Understanding statistical concepts like the standard normal distribution provides the foundation for further study and application in more complex statistical analyses. With effective statistics education, students become well-versed in interpreting patterns and making evidence-based decisions.

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    Most popular questions from this chapter

    Determine whether you can use a normal distribution to approximate the binomial distribution. If you can, use the normal distribution to approximate the indicated probabilities and sketch their graphs. If you cannot, explain why and use a binomial distribution to find the indicated probabilities. Sixty-five percent of U.S. college graduates are employed in their field of study. You randomly select 20 U.S. college graduates and ask them whether they are employed in their field of study. Find the probability that the number who are employed in their field of study is (a) exactly 15 , (b) less than 10 , and (c) between 20 and 35 . Identify any unusual events. Explain. (Source: Accenture)

    A population and sample size are given. (a) Find the mean and standard deviation of the population. (b) List all samples (with replacement) of the given size from the population and find the mean of each. (c) Find the mean and standard deviation of the sampling distribution of sample means and compare them with the mean and standard deviation of the population. The goals scored in a season by the four starting defenders on a soccer team are \(1,2,0\), and \(3 .\) Use a sample size of \(2 .\)

    Find the indicated area under the standard normal curve. If convenient, use technology to find the area. To the left of \(z=-0.11\)

    Find the indicated probabilities and interpret the results. The mean annual salary for intermediate level life insurance underwriters is about \(\$ 61,000 .\) A random sample of 45 intermediate level life insurance underwriters is selected. What is the probability that the mean annual salary of the sample is (a) less than \(\$ 60,000\) and (b) more than \(\$ 63,000 ?\) Assume \(\sigma=\$ 11,000\). (Adapted from Salary.com)

    Find the indicated probability using the standard normal distribution. If convenient, use technology to find the probability. $$P(z>-0.74)$$
    See all solutions

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