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Chapter 5: Problem 39

Use the Standard Normal Table or technology to find the \(z\)-score that corresponds to the cumulative area or percentile. $$0.993$$

Short Answer

Expert verified
The \(z\)-score is approximately \(2.47\).

Step by step solution

01

Understanding the Problem

We need to find the \(z\)-score for which the cumulative area to the left of \(z\) is \(0.993\). This means we are looking for the \(z\)-score such that \(P(Z \leq z) = 0.993\) in a standard normal distribution.
02

Locate Cumulative Area in Table

Using the Standard Normal Table, find the value closest to \(0.993\). The table lists cumulative probabilities from the left.
03

Identify the Closest \(z\)-Score

In the Standard Normal Table, the closest cumulative probability to \(0.993\) is \(0.9931\), which corresponds to a \(z\)-score of \(2.47\).
04

Verification with Technology

Alternatively, use a calculator or statistical software to confirm the \(z\)-score by computing \(z = \Phi^{-1}(0.993)\), where \(\Phi\) is the cumulative distribution function of the standard normal distribution.

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Key Concepts

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

Z-Score Calculation
A z-score tells us how many standard deviations away a particular data point is from the mean of a distribution. To calculate a z-score, you can use the formula:
  • \( z = \frac{(X - \mu)}{\sigma} \)
  • where \( X \) is the value you are assessing, \( \mu \) is the mean of the distribution, and \( \sigma \) is the standard deviation.
In a standard normal distribution, the mean \( \mu \) is 0 and the standard deviation \( \sigma \) is 1. Therefore, the z-score simplifies to:
  • \( z = X \)
Z-scores are useful because they allow us to determine the probability of a value occurring within a normal distribution. Finding the z-score for a cumulative probability involves identifying the z-score associated with a given probability value, which can be done using a z-table or statistical software. In our problem, we are finding the z-score for a cumulative area of 0.993.
Cumulative Probability
Cumulative probability refers to the likelihood that a random variable is less than or equal to a certain value. In the context of a normal distribution, cumulative probability helps us understand the proportion of data points on the left of a specific point on the curve.
  • In practical terms, it shows how much of the distribution falls below a certain point.
  • For instance, a cumulative probability of 0.993 implies that 99.3% of the data is below the given z-score.
This concept allows us to make predictions and decisions based on statistical data, as it offers a comprehensive view of where a specific value stands relative to the entire distribution. It is particularly useful when working with larger datasets.
Normal Distribution
The normal distribution, also known as the Gaussian distribution, is a continuous probability distribution that is symmetrical around its mean. It is often depicted as a bell-shaped curve and is characterized by its mean and standard deviation. Here are some key features:
  • The mean, median, and mode of a normal distribution are equal.
  • Approximately 68% of the data falls within one standard deviation of the mean, 95% within two, and 99.7% within three.
This distribution is fundamental in statistics because many natural phenomena and measurement errors tend to follow a normal distribution. In our exercise, we are specifically dealing with the standard normal distribution, which is a special case where the mean is 0 and the standard deviation is 1. Using this standardization makes it easier to calculate cumulative probabilities and find corresponding z-scores.
Statistical Software
Statistical software is a vital tool in analyzing and interpreting data, particularly when working with large datasets or complex calculations. These programs can perform various statistical tests and calculations quickly and accurately, freeing analysts from tedious manual computations.
  • Popular software includes SPSS, R, SAS, and Python's SciPy library.
  • These programs typically come equipped with functions to compute z-scores, cumulative probabilities, and inverse functions for distribution tables.
  • For example, to find the z-score that corresponds to a cumulative probability in Python, the SciPy library offers the `ppf` function, which computes the inverse of the cumulative distribution function.
Statistical software is especially beneficial for verifying the results obtained from z-tables or when the exact values do not appear in the tables. For accuracy and efficiency, it's a valuable addition to any statistician's toolkit.

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Most popular questions from this chapter

The random variable \(x\) is normally distributed with mean \(\mu=74\) and standard deviation \(\sigma=8\). Find the indicated probability. $$P(x<57)$$

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 \(z\)-score that has \(20.9 \%\) of the distribution's area to its right.

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. A survey of U.S. adults found that \(32 \%\) have an online account with their healthcare provider. You randomly select 70 U.S. adults and ask them whether they have an online account with their healthcare provider. Find the probability that the number who have an online account with their healthcare provider is (a) at most 15 , (b) exactly 25 , and (c) greater than 30 . Identify any unusual events. Explain. (Source: Pew Research Center)

Use the Standard Normal Table or technology to find the \(z\)-score that corresponds to the cumulative area or percentile. $$0.4364$$
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