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Chapter 6: Problem 1

Given a standard normal distribution. find the normal curve area under the curve which lies (a) to the left of \(z=1.43\); (b) to the right of \(z=-0.89\) : (c) between \(z=-2.16\) and \(z=-0.65\) (el) to the left of; \(=-1.39\); (e) to the: right of \(z=1.96\) : (T) between \(z=-0.48\) and \(z=1.74\).

Short Answer

Expert verified
The areas under the curve for the standard normal distribution are: (a) 0.9236, (b) 0.8133, (c) 0.2424, (d) 0.0823. (e) 0.0250 , (f) 0.6435.

Step by step solution

01

Area to the left of \(z=1.43\)

Looking up \(1.43\) in the standard normal distribution table gives a value of \(0.9236\). Therefore, the area under the curve to the left of \(z = 1.43\) is \(0.9236\).
02

Area to the right of \(z=-0.89\)

Looking up \(-0.89\) in the table gives \(0.1867\). This is the area to the left of \(z=-0.89\). To find the area to the right, subtract this value from 1, i.e., \(1 - 0.1867 = 0.8133\).
03

Area between \(z=-2.16\) and \(z=-0.65\)

Starting with \(z=-2.16\), the area to the left is \(0.0154\). For \(z=-0.65\), the area to the left is \(0.2578\). To find the area between these two values, subtract the two areas: \(0.2578 - 0.0154 = 0.2424\).
04

Area to the left of \(z=-1.39\)

Looking up \(-1.39\) in the standard normal distribution table yields a value of \(0.0823\). Therefore, the area under the curve to the left of \(z=-1.39\) is \(0.0823\).
05

Area to the right of \(z=1.96\)

Looking up \(1.96\) in the standard normal distribution table gives a value of \(0.9750\). To find the area to the right, subtract this value from 1: \(1 - 0.9750 = 0.0250\).
06

Area between \(z=-0.48\) and \(z=1.74\)

The area to the left of \(z=-0.48\) is \(0.3156\), and the area to the left of \(z=1.74\) is \(0.9591\). To find the area between these two z-scores, subtract the corresponding areas: \(0.9591 - 0.3156 = 0.6435\).

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Key Concepts

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

Standard Normal Table
The standard normal table, also known as the z-table, is a crucial tool when working with normal distributions. It provides the areas under the curve of a normal distribution for any given z-score. This applies to a situation where the mean is zero and the standard deviation is one. In simpler terms, it's like a map that helps us find the probability of data points falling below a certain point in a standard normal distribution.

The table is typically used in two ways:
  • To find the probability to the left of a specific z-score.
  • To calculate the probability associated with a given z-score interval.
To use the table, locate the row corresponding to the z-score's first decimal and the column corresponding to its second decimal. The intersection gives you the area to the left of that z-score.
Z-scores
Z-scores are standardized scores that indicate how many standard deviations a data point is from the mean of the dataset. In the context of normal distribution, they allow us to understand where a particular value lies relative to the average.

Z-scores are calculated using the formula:\[ z = \frac{(X - \mu)}{\sigma} \]Where:
  • \(X\) is the value,
  • \(\mu\) is the mean of the data set,
  • \(\sigma\) is the standard deviation.
These scores are essential because they allow for the comparison of scores from different distributions by putting them on a common scale.
Area under the Curve
The area under a curve in a normal distribution graph represents the total probability of outcomes within a given range. When we refer to the area under the curve in the context of z-scores, we are talking about the cumulative probability up to that point.

For example, if you're interested in a specific range of z-scores, the area under the curve represents the probability that a data point falls within that range. It can be adapted based on where the z-scores lie:
  • Area to the left of a z-score: Directly read from the standard normal table.
  • Area to the right: Calculated by subtracting the table value from 1.
  • Area between two z-scores: Difference between the two table values.
Understanding this concept is vital as it is the foundation for calculating probabilities in a normal distribution.
Probability Calculations
Probability calculations in a normal distribution are fundamental to inferential statistics, helping us make predictions about a dataset. Using z-scores and the standard normal table, we can easily determine the likelihood of a score falling within a certain range or beyond certain limits.

Here's how it works:
  • Calculate the z-score for the data value if it isn't given.
  • Use the standard normal table to find areas corresponding to the z-scores, which indicate probabilities.
  • Apply the appropriate method for the type of area calculation (left, right, or between two points).
These steps allow you to answer questions like "What is the chance of scoring above a certain value?" or "How likely is a value to fall between two points?" Mastery of these calculations is crucial for accurate data analysis and decision-making in statistics.

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Most popular questions from this chapter

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