Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 6: Problem 18
Obtain the area under the standard normal curve a. to the right of \(z=1.43\) b. to the left of \(z=-1.65\) c. to the right of \(z=-.65\) d. to the left of \(z=.89\)
Short Answer
Expert verified
The areas under the curve are a. 0.0764, b. 0.0495, c. 0.2578, and d. 0.8133.
Step by step solution
01
Understanding the Z-score
The z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It鈥檚 measured in terms of standard deviations from the mean. For a normal distribution, it determines the spread of distribution.
02
Using the standard normal distribution table
To calculate the area under the curve, use a standard normal distribution table. This table provides the probability that a statistic is observed below, above, or between values. The value of the table is the area to the left of a particular z-score.
03
Solve for (a) Right of \(z=1.43\)
Look up \(z=1.43\) in the standard normal table. This gives the area to the left of \(z=1.43\) as 0.9236. Since the total area under the curve is 1, the area to the right of \(z=1.43\) is \(1 - 0.9236 = 0.0764\)
04
Solve for (b) Left of \(z=-1.65\)
Look up \(z=1.65\) in the standard normal table. The area to the left of \(z=-1.65\) is the same as the area to the right of \(z=1.65\). From the table, the area to the left of \(z=1.65\) is 0.9505, so the area to the right of \(z=1.65\) is \(1 - 0.9505 = 0.0495\). Thus, the area to the left of \(z=-1.65\) is 0.0495.
05
Solve for (c) Right of \(z=-0.65\)
The area to the right of \(z=-0.65\) is the same as the area to the left of \(z=0.65\). Look up \(z=0.65\) in the standard normal table to find an area of 0.7422. The area to the right of \(z=-0.65\) is therefore 1 - 0.7422 = 0.2578.
06
Solve for (d) Left of \(z=0.89\)
Look for \(z=0.89\) in the standard normal table. The area to the left of \(z=0.89\) is 0.8133 (from the table).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
z-score
The z-score is an important concept in statistics that helps us understand how far away a particular value is from the mean of a distribution. It tells us how many standard deviations a value is from the mean. If you have a positive z-score, it means your value is above the mean. Conversely, a negative z-score indicates a value below the mean.
Understanding z-scores allows us to place our data in context relative to a standard normal distribution. When we know the z-score, we can easily find areas under the curve, which reflects probabilities or proportions of the data.
Understanding z-scores allows us to place our data in context relative to a standard normal distribution. When we know the z-score, we can easily find areas under the curve, which reflects probabilities or proportions of the data.
- The formula for calculating a z-score is: \[ z = \frac{(X - \mu)}{\sigma} \]where \( X \) is the value in question, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.
- Z-scores help compare individual data points to a model of standard normal distribution.
- A z-score of 0 means the data point is exactly at the mean.
normal distribution table
The normal distribution table, also known as the z-table, is used to find the area (or probability) associated with different z-scores. This comes in handy when working with a standard normal distribution, which is a bell-shaped curve that is symmetrical around its mean.
The table shows the cumulative probability of a normal distribution up to a given z-score. It gives the cumulative probability from the left up to a particular z-score. To calculate different probabilities:
The table shows the cumulative probability of a normal distribution up to a given z-score. It gives the cumulative probability from the left up to a particular z-score. To calculate different probabilities:
- To find the probability to the right of a z-score, subtract the table value from 1.
- The table is usually organized with the z-score's integer and first decimal on the left, while the second decimal place is on top.
- Z-tables are essential for determining percentiles, probabilities, and for performing hypothesis tests.
area under the curve
The area under the curve in a standard normal distribution represents the probability of a certain range of data occurring. The total area under the curve is always equal to 1, reflecting the total probability.
Here's how areas under the curve work:
Here's how areas under the curve work:
- The area to the left of a given z-value provides the probability of a randomly selected value being less, or equal, to that z-score.
- When looking for the probability of a value occurring to the right of a z-score, the equation used is:\[ P(Z > z) = 1 - P(Z < z) \]
- Areas give insights into statistical significance and help make inferences about data sets.
- This has real-world applications like finding how unusual or normal a particular result may be within its distribution context.
statistical measurement
Statistical measurement is a component of statistics that deals with the collection, analysis, and interpretation of data. In the context of z-scores and normal distribution, it focuses on determining the likelihood and behavior of data under the curve.
Z-scores and areas under the curve are tools in statistical measurement to :
Z-scores and areas under the curve are tools in statistical measurement to :
- Determine whether a data point lies within the typical range of a population.
- Evaluate hypotheses in research by examining z-scores within the context of a distribution.
- Offer a way to standardize and compare datasets even when they're measured in different units.
- Provide a deeper understanding of variability and distribution within datasets.
