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Chapter 6: Problem 24

Sketch the areas under the standard normal curve over the indicated intervals, and find the specified areas. $$ \text { Between } z=-1.40 \text { and } z=2.03 $$

Short Answer

Expert verified
The area between \( z = -1.40 \) and \( z = 2.03 \) is 0.8980.

Step by step solution

01

Understand the problem

We need to find the area under the standard normal distribution curve between the z-scores of -1.40 and 2.03. This area corresponds to the probability of a standard normal variable falling within this interval.
02

Identify the relevant probability

The area under the standard normal distribution curve between two z-scores, say \( z_1 \) and \( z_2 \), represents the probability \( P(z_1 < Z < z_2) \). Here, \( z_1 = -1.40 \) and \( z_2 = 2.03 \).
03

Use the z-table

Look up the cumulative probability for \( z = -1.40 \) and \( z = 2.03 \) in the standard normal distribution table (z-table). \( P(Z < -1.40) \) corresponds to 0.0808 and \( P(Z < 2.03) \) corresponds to 0.9788.
04

Calculate the area between the z-scores

Subtract the cumulative probability at \( z = -1.40 \) from the cumulative probability at \( z = 2.03 \). The area between \( z = -1.40 \) and \( z = 2.03 \) is \( P(-1.40 < Z < 2.03) = 0.9788 - 0.0808 = 0.8980 \).
05

Sketch the area under the curve

Draw the standard normal curve, a smooth bell-shaped curve centered at zero. Mark the z-scores \( z = -1.40 \) and \( z = 2.03 \) on the horizontal axis, shading the area between them to represent the probability calculated in the previous step.

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Key Concepts

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

Z-scores
Z-scores are a way to measure how far away a given data point is from the mean of a data set, expressed in terms of standard deviations. Essentially, a z-score tells us how unusual or typical a value is within a normal distribution.
Understanding z-scores can be simplified as follows:
  • A z-score of 0 indicates that the data point is exactly at the mean.
  • Positive z-scores indicate a value above the mean, while negative z-scores indicate a value below the mean.
  • Larger absolute values of z-scores (whether positive or negative) represent values further from the mean, often referred to as outliers.
The formula for calculating a z-score is: \[ z = \frac{X - \mu}{\sigma} \]where:
  • \( X \) is the value of the element.
  • \( \mu \) is the mean of the data set.
  • \( \sigma \) is the standard deviation of the data set.
Recognizing z-scores is crucial when dealing with normally distributed data and helpful in making statistical inferences.
Probability
Probability, at its core, is the measure of the likelihood that a certain event will occur. It ranges from 0 to 1, where 0 indicates impossibility and 1 indicates certainty. In the context of the standard normal distribution, probability is represented by the area under the normal curve.
There are a few key points about probability to remember:
  • When dealing with a normal distribution, probabilities correspond to areas under the curve.
  • The total area under the standard normal distribution curve is 1, representing complete certainty.
  • The probability of an event occurring between two z-scores is the area under the curve between those z-scores.
Calculating probabilities using the standard normal distribution often involves z-scores and cumulative probability tables, commonly referred to as z-tables.
Normal Curve
The normal curve, often referred to as the bell curve, is a graphical representation of the normal distribution. It is symmetrical around the mean, depicting how data is dispersed.
The properties of the normal curve include:
  • It is symmetric around the mean.
  • It has a single peak at the mean which is also its median and mode.
  • The curve is asymptotic, meaning it approaches the axis but never touches it.
  • The area under the entire curve adds up to 1 (or 100% probability).
The standard normal curve, specifically, is a special type of normal curve where the mean (\( \mu \)) is 0 and the standard deviation (\( \sigma \)) is 1.
Visualizing data as a normal curve helps in understanding the distribution of data points and their likelihoods.
Cumulative Probability
Cumulative probability is the probability that a random variable will take a value less than or equal to a specific value. In the realm of standard normal distribution, this is calculated using z-tables.
Here's how cumulative probability works:
  • These probabilities accumulate over an interval and can be found in z-tables, which provide the area from the mean to a specific z-score.
  • The cumulative probability up to a certain z-score reflects the total area under the curve up to that score.
  • It helps in determining the likelihood of a value falling below a particular point in the dataset.
For instance, in our scenario:
  • The cumulative probability at \( z = -1.40 \) is 0.0808.
  • The cumulative probability at \( z = 2.03 \) is 0.9788.
The difference between these cumulative probabilities gives the probability of a value falling between \( z = -1.40 \) and \( z = 2.03 \).
This area represents the likelihood, helping us draw conclusions about dataset positioning and probability scenarios.

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Most popular questions from this chapter

Find the \(z\) value described and sketch the area described. Find \(z\) such that \(97.5 \%\) of the standard normal curve lies to the left of \(z\).

Assume that \(x\) has a normal distribution with the specified mean and standard deviation. Find the indicated probabilities. $$ P(7 \leq x \leq 9) ; \mu=5 ; \sigma=1.2 $$

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

More than a decade ago, high levels of lead in the blood put \(88 \%\) of children at risk. A concerted effort was made to remove lead from the environment. Now, according to the Third National Health and Nutrition Examination Survey (NHANES III) conducted by the Centers for Disease Control, only \(9 \%\) of children in the United States are at risk of high blood-lead levels. (a) In a random sample of 200 children taken more than a decade ago, what is the probability that 50 or more had high blood-lead levels? (b) In a random sample of 200 children taken now, what is the probability that 50 or more have high blood-lead levels?

The college Physical Education Department offered an Advanced First Aid course last semester. The scores on the comprehensive final exam were normally distributed, and the \(z\) scores for some of the students are shown below: \(\begin{array}{lll}\text { Robert, } 1.10 & \text { Juan, } 1.70 & \text { Susan, }-2.00\end{array}\) \(\begin{array}{ll}\text { Joel, } 0.00 & \text { Jan, }-0.80\end{array}\) Linda, \(1.60\) (a) Which of these students scored above the mean? (b) Which of these students scored on the mean? (c) Which of these students scored below the mean? (d) If the mean score was \(\mu=150\) with standard deviation \(\sigma=20\), what was the final exam score for each student?
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