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Chapter 6: Problem 19

Find the \(z\) value described and sketch the area described. Find \(z\) such that \(8 \%\) of the standard normal curve lies to the right of \(z\).

Short Answer

Expert verified
The z-score is 1.41.

Step by step solution

01

Understanding the Problem

We need to find the z-score such that only 8% of the standard normal distribution lies to the right of this z-score. This means that the total area to the right of the z-score is 0.08.
02

Finding the Area to the Left

Since the total area under the standard normal curve is 1, the area to the left of the z-score is calculated as 1 - 0.08 = 0.92.
03

Using the Z-Table

Look up the value 0.92 in the standard normal distribution table (Z-table), which shows cumulative probabilities from the left. Find the nearest value to 0.92 and locate its corresponding z-score.
04

Interpreting the Z-Table Value

For 0.92 cumulative probability, the closest z-score is approximately 1.41. This is determined by finding the closest value in the Z-table and reading the corresponding z-score from the row and column headers.
05

Sketching the Area

Draw the standard normal curve (bell curve) and mark a vertical line at z = 1.41. Shade the area to the right of this line, which represents 8% of the total area under the 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, also known as a standard score, is a statistic that tells us how many standard deviations an element is from the mean of a distribution. In simpler terms, it measures the position of a value within a normal distribution.
  • If a Z-score is 0, it means the data point is exactly average, located at the mean.
  • A positive Z-score indicates that the data point is above the mean.
  • A negative Z-score signals that it is below the mean.
In the context of our exercise, finding the z-score where 8% lies to its right involves understanding these standard deviations along a normal curve. The ability to interpret a Z-score helps in determining probabilities and making statistical inferences.
Cumulative Probability
Cumulative probability refers to the likelihood that a random variable is less than or equal to a certain value. It is calculated by summing up individual probabilities of outcomes up to that specific point. In standardized terms with the standard normal distribution:
  • The cumulative probability is calculated from the left up to the specific Z-score.
  • Since the total area under the curve is 1 (representing 100% probability), the cumulative probability is the area under the curve up to a certain z-value.
In our problem, we translate the 8% area to the right by calculating the cumulative probability to the left: 1 - 0.08 = 0.92. This means 92% of the data falls below this particular z-score.
Z-table
The Z-table, or standard normal distribution table, is a chart that displays cumulative probability values associated with each possible Z-score.
  • It helps find the probability that a statistic is less than a specified Z value, or vice versa.
  • The column and row headers of the Z-table dictate specific Z-scores for cumulative probabilities given in the table.
In reference to the problem, to find a z-score corresponding to a cumulative probability of 0.92, one must locate 0.92 within the Z-table and then read the corresponding Z value. In this instance, we find that for 0.92 probability, the Z-score is approximately 1.41. This indicates that this z-score is more than one standard deviation above the mean.
Normal Curve
The normal curve, or bell curve, is a graphical representation of a normal distribution. It is symmetrical, with most of the observations clustering around the central peak or mean. Key attributes of a normal curve:
  • The peak of the curve represents the mean, median, and mode, which are all the same in a perfectly normal distribution.
  • The curve spreads out from the mean, with its shape governed by the standard deviation.
  • As one moves away from the center, the probability of observations decreases exponentially.
In exercises like ours, shading areas under the normal curve helps visually represent probabilities, such as marking 8% of the curve's area to the right of the z-score. It provides a clear visual interpretation of the standard deviations and probabilities associated with normal distributions.

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Most popular questions from this chapter

Suppose \(5 \%\) of the area under the standard normal curve lies to the left of \(z\). Is \(z\) positive or negative?

What does a standard score measure?

Let \(z\) be a random variable with a standard normal distribution. Find the indicated probability, and shade the corresponding area under the standard normal curve. $$ P(-0.45 \leq z \leq 2.73) $$

Let \(z\) be a random variable with a standard normal distribution. Find the indicated probability, and shade the corresponding area under the standard normal curve. $$ P(-2.18 \leq z \leq-0.42) $$

Assume that \(x\) has a normal distribution with the specified mean and standard deviation. Find the indicated probabilities. $$ P(x \geq 90) ; \mu=100 ; \sigma=15 $$
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