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Chapter 3: Problem 29

Standard Normal drill. (a) Find the number \(z\) such that the proportion of observations that are less than \(z\) in a standard Normal distribution is \(0.2\). (b) Find the number \(z\) such that \(40 \%\) of all observations from a standard Normal distribution are greater than z.

Short Answer

Expert verified
(a) \(-0.84\), (b) \(0.25\)

Step by step solution

01

Identify the Value for Part (a)

We are asked to find the z-value such that 20% of the observations are less than this value. In a standard normal distribution, this means we need to find the z-score that corresponds to a cumulative probability of 0.2.
02

Determine the z-score for 0.2

Using a standard normal distribution table or a calculator, we look for the z-score where the cumulative probability is closest to 0.2. This corresponds to a z-score of approximately \(-0.84\).
03

Interpret Part (b)

For part (b), we need the z-value where 40% of the observations are greater than this value. This means we are looking for the z-score that leaves 60% of the observations below it (since 100% - 40% = 60%).
04

Find the z-score for 0.6

Referring to the standard normal distribution table or a calculator, we find the z-score that corresponds to a cumulative probability of 0.6. This is approximately \(0.25\).

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Key Concepts

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

Understanding the Z-Score
The z-score plays an integral role in statistics, particularly when dealing with standard normal distribution. It essentially measures how many standard deviations an element is from the mean. In a standard normal distribution, the mean is 0 and the standard deviation is 1.
  • A z-score of 0 indicates that the data point's score is identical to the mean score.
  • A positive z-score indicates a value above the mean, while a negative one indicates a value below the mean.
Understanding the z-score is crucial because it allows statisticians to understand where a specific observation falls in relation to the majority of data values in a normally distributed dataset.
Exploring Cumulative Probability
Cumulative probability refers to the probability that a random variable will be less than or equal to a certain value. In the context of the standard normal distribution, it helps to determine how much of the distribution lies to the left of a particular z-score.
  • Cumulative probabilities are expressed in decimal form, representing a percentage of the total distribution.
  • For instance, if a z-score has a cumulative probability of 0.2, it means that 20% of the data lies below that z-score.
  • Similarly, if a z-score has a cumulative probability of 0.6, 60% of the data lies below this value, leaving 40% above it.
Cumulative probability is a helpful concept in statistics problem-solving as it allows for the determination of the distribution of data relative to particular z-scores.
Approaching Statistics Problem-Solving
Statistics problem-solving often involves a methodical approach to interpret and analyze data. Tasks like identifying z-scores, as shown in the original exercise, require understanding both the theoretical and practical application of statistical concepts. Here's a suggested approach:
  • Identify the Problem: Clearly determine what is being asked鈥攊n this case, the z-score corresponding to a given cumulative probability.
  • Gather Information: Use the standard normal distribution table or a calculator to find the necessary z-scores.
  • Interpret Results: Translate the numerical findings into meaningful insights regarding your dataset.
By breaking down problems into these steps, complex statistics problems become more manageable and less intimidating.
Using the Normal Distribution Table
A normal distribution table, often referred to as the z-table, is an essential tool in statistics. It helps find the probability that a standard normal random variable is less than or equal to a given z-score. Understanding how to read it can greatly enhance your ability to solve problems involving standard normal distribution.
  • The z-table provides cumulative probabilities for z-scores, typically covering only half from 0 to positive infinity because of the symmetry of the normal distribution.
  • To find a z-score for a particular cumulative probability, locate the probability in the table. Then, follow the corresponding row and column, which gives you the z-score.
  • The use of a normal distribution table is prevalent in statistics due to its ability to provide quick and accurate results for probabilities related to the normal distribution.
Remember, the table is a powerful aid, but understanding how to use it efficiently is key to solving statistics problems accurately.

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Most popular questions from this chapter

To completely specify the shape of a Normal distribution, you must give (a) the mean and the standard deviation. (b) the five-number summary. (c) the median and the quartiles.

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