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Chapter 6: Problem 14

Find the area under the standard normal curve a. to the right of \(z=1.36\) b. to the left of \(z=-1.97\) c. to the right of \(z=-2.05\) d. to the left of \(z=1.76\)

Short Answer

Expert verified
Without an appropriate standard normal distribution table, it's not possible to give numerical answers. Though, for each case, the correct answer would be the numeric value that results from the operations mentioned above, given a standard z-table or calculator is used.

Step by step solution

01

- Understand the standard normal distribution curve and z-scores

The standard normal distribution, also referred to as the z-distribution, is a special normal distribution with a mean of 0 and standard deviation of 1. The z-score tells us how many standard deviations an element is from the mean. The area under the curve correspond to probabilities and the total area under the curve is always equal to 1.
02

- Use of Z-tables

After understanding the basics, we will use a standard z-table or appropriate statistical software to find the area under the standard normal curve for the given z-values. A Z-table provides the probabilities for the standard normal distribution. It shows the probabilities up the given z-score, so for finding the probabilities to the right of the z-value, we have to subtract the table value from 1.
03

- Find the area under the curve for each case

a. To find the area to the right of \(z=1.36\), we look up this z-score in the z-table and subtract from 1 because we're interested in the area to the right -(1-P(Z<1.36))=b. For \(z=-1.97\), we directly take the complement from the table because it's to the left - P(Z<-1.97).c. For \(z=-2.05\), the area to the right is 1-P(Z<-2.05)d. For \(z=1.76\), look up this z-score in the z-table - P(Z<1.76)

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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 key part of understanding the standard normal distribution. In simple terms, a z-score tells us how many standard deviations away a particular point is from the mean of the distribution. The standard normal distribution has a mean of 0 and a standard deviation of 1.
  • If a z-score is positive, it means the data point is above the mean.
  • If it's negative, the point is below the mean.
Understanding z-scores is crucial because they allow us to determine the position of a value within a distribution. By calculating z-scores for different values, we can easily compare scores from different normal distributions.
z-table
A z-table, also known as a standard normal distribution table, is a tool used to find the probability of a statistic under the normal distribution curve. It contains the probabilities of a z-score falling to the right or the left of the mean. This table essentially takes a z-score and gives you a probability value.

When using a z-table:
  • Locate the z-score on the table, typically by finding the row for the ten's and one's digit and the column for the hundredth's digit.
  • The value at the intersection gives you the cumulative probability up to that z-score.
  • For right-side probabilities, subtract the table value from 1, since standard z-tables usually provide left-side probabilities.
Knowing how to use a z-table is essential for working effectively with normal distribution problems.
area under the curve
The area under a normal distribution curve is vital because it represents probabilities. The total area under the curve is always 1, corresponding to a 100% probability. This concept is used to find probabilities of a z-score occurring within a certain range.
  • To find probabilities to the right of a z-score, calculate 1 minus the area to the left.
  • For left probabilities, directly use the area provided by the z-table.
These areas are helpful not only in determining probabilities but also in making statistical inferences about a population based on sample data. The concept of area under the curve underlies much of inferential statistics and decision making.
probability
Probability in the context of the normal distribution refers to the likelihood of an event occurring within a given range of values. This is represented by areas under the standard normal curve. In a normal distribution, this is directly related to z-scores and the z-table values.
  • Probabilities range from 0 to 1, where 0 indicates impossibility and 1 indicates certainty.
  • The probability tied to a z-score gives the chance of finding a data point at or below that score.
Understanding probabilities in terms of the area under the curve allows us to make predictions and decisions based on statistical data. It is a fundamental concept in statistics that helps in estimating outcomes and risks. For example, determining the likelihood of a student's test score falling within a certain range can be calculated using these probabilistic methods.

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Most popular questions from this chapter

What is the difference between the probability distribution of a discrete random variable and that of a continuous random variable? Explain.

Let \(x\) be a continuous random variable that follows a normal distribution with a mean of 550 and a standard deviation of \(75 .\) a. Find the value of \(x\) so that the area under the normal curve to the left of \(x\) is \(.0250\). b. Find the value of \(x\) so that the area under the normal curve to the right of \(x\) is \(.9345\). c. Find the value of \(x\) so that the area under the normal curve to the right of \(x\) is approximately .0275. d. Find the value of \(x\) so that the area under the normal curve to the left of \(x\) is approximately \(.9600\). e. Find the value of \(x\) so that the area under the normal curve between \(\mu\) and \(x\) is approximately \(.4700\) and the value of \(x\) is less than \(\mu\). f. Find the value of \(x\) so that the area under the normal curve between \(\mu\) and \(x\) is approximately \(.4100\) and the value of \(x\) is greater than \(\mu\).

A construction zone on a highway has a posted speed limit of 40 miles per hour. The speeds of vehicles passing through this construction zone are normally distributed with a mean of 46 miles per hour and a standard deviation of 4 miles per hour. Find the percentage of vehicles passing through this construction zone that are a. exceeding the posted speed limit b. traveling at speeds between 50 and 57 miles per hour

At Jen and Perry Ice Cream Company, a machine fills 1 -pound cartons of Top Flavor ice cream. The machine can be set to dispense, on average, any amount of ice cream into these cartons. However, the machine does not put exactly the same amount of ice cream into each carton; it varies from carton to carton. It is known that the amount of ice cream put into each such carton has a normal distribution with a standard deviation of \(.18\) ounce. The quality control inspector wants to set the machine such that at least \(90 \%\) of the cartons have more than 16 ounces of ice cream. What should be the mean amount of ice cream put into these cartons by this machine?

The Jen and Perry Ice Cream company makes a gourmet ice cream. Although the law allows ice cream to contain up to \(50 \%\) air, this product is designed to contain only \(20 \%\) air. Because of variability inherent in the manufacturing process, management is satisfied if each pint contains between \(18 \%\) and \(22 \%\) air. Currently two of Jen and Perry's plants are making gourmet ice cream. At Plant A, the mean amount of air per pint is \(20 \%\) with a standard deviation of \(2 \%\). At Plant \(\mathrm{B}\), the mean amount of air per pint is \(19 \%\) with a standard deviation of \(1 \%\). Assuming the amount of air is normally distributed at both plants, which plant is producing the greater proportion of pints that contain between \(18 \%\) and \(22 \%\) air?
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