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Chapter 4: Problem 2

Area under the curve, Part II. What percent of a standard normal distribution \(N(\mu=0, \sigma=1)\) is found in each region? Be sure to draw a graph. (a) \(Z>-1.13\) (b) \(Z<0.18\) (c) \(Z>8\) (d) \(|Z|<0.5\)

Short Answer

Expert verified
(a) 87.08%, (b) 57.14%, (c) 0%, (d) 38.30%.

Step by step solution

01

Understanding the Standard Normal Distribution

A standard normal distribution, denoted as \(N(\mu=0, \sigma=1)\), is a normal distribution with a mean \(\mu\) of 0 and a standard deviation \(\sigma\) of 1. It is represented by the bell-shaped curve where total area under the curve equals 1.
02

Using the Z-table or Z-chart

The Z-table or Z-chart helps us find the area (or probability) under the curve to the left of a given Z-score.
03

Calculating Area for (a) Z > -1.13

Use the Z-table to find the area to the left of \(Z = -1.13\). The table gives approximately 0.1292. The probability for \(Z > -1.13\) is 1 minus this value: \(1 - 0.1292 = 0.8708\) or 87.08%.
04

Calculating Area for (b) Z < 0.18

Look up \(Z = 0.18\) in the Z-table, which gives approximately 0.5714. This means the probability for \(Z < 0.18\) is 57.14%.
05

Calculating Area for (c) Z > 8

In a standard normal distribution, a Z-score of 8 is extremely far in the tails. The area under the curve at this point is practically 0, implying \(P(Z > 8)\) is approximately 0%.
06

Calculating Area for (d) |Z| < 0.5

This is the same as finding the area for \(-0.5 < Z < 0.5\). Look up these Z-values: \(Z = 0.5\) gives about 0.6915 and \(Z = -0.5\) gives about 0.3085. The area between them is \(0.6915 - 0.3085 = 0.3830\) or 38.30%.

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Key Concepts

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

Z-score
The Z-score is a measure of how many standard deviations an element is from the mean in a standard normal distribution. This score helps in understanding where a particular value fits within the distribution. If you have a Z-score of 0, it means that your value is exactly the average.
  • A positive Z-score indicates a value above the mean.
  • A negative Z-score shows a value below the mean.
To calculate the Z-score, you can use the formula:\[Z = \frac{(X - \mu)}{\sigma}\]where \(X\) is the value in question, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. Understanding the Z-score is essential for finding probabilities and interpreting data points within the distribution.
probability
In statistics, probability is the likelihood or chance of an event occurring. When dealing with a standard normal distribution, you often use the Z-score to determine probabilities, or areas under the curve. The total area under the standard normal distribution curve is 1, representing the probability of all outcomes in the distribution.
When you want to find out the probability of a certain event, you're looking at the area under the curve for a certain range:
  • For example, if you want the probability of a event with \(Z > -1.13\), you would calculate this by subtracting the cumulative probability up to \(-1.13\) from 1.
  • In contrast, for something like \(|Z| < 0.5\), you find the range between \(-0.5\) and \(0.5\) and look at the area between those Z-scores.
Understanding these probabilities is crucial for areas such as hypothesis testing and determining confidence intervals.
Z-table
The Z-table, also known as the standard normal table, is a mathematical chart that allows you to find probabilities associated with a specific Z-score. It helps you understand the likelihood of a particular value occurring under the bell-shaped curve of a standard normal distribution.
Here's how you use a Z-table:
  • Locate your Z-score in the table; Z-tables typically list values for positive and negative Z-scores separately.
  • Find the corresponding area value, which represents the cumulative probability from the left up to that Z-score.
For instance, if you have a Z-score of 0.18, the Z-table might show you an area of 0.5714, meaning there's a 57.14% chance that a value lies below this Z-score. Z-tables are vital tools when working with normal distributions, especially in calculating probabilities and understanding data behavior.
normal distribution
The normal distribution, often called a bell curve due to its shape, is a key concept in probability and statistics. It describes how data is spread around a mean (\(\mu\)) with a certain standard deviation (\(\sigma\)). In a standard normal distribution, the mean is 0, and the standard deviation is 1.
Key features of a normal distribution include:
  • Symmetrical around the mean, meaning most data points are close to the average.
  • As you move further from the mean, the probability of data points decreases rapidly.
  • Total area under the curve is equal to 1, accounting for all possible outcomes.
Examples include the heights of people, test scores, or measurement errors; a standard normal distribution allows us to calculate and interpret probabilities easily. It's fundamental in various fields, including finance, natural sciences, and social sciences, and is a starting point for statistical hypothesis testing.

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Most popular questions from this chapter

Defective rate. A machine that produces a special type of transistor (a component of computers) has a \(2 \%\) defective rate. The production is considered a random process where each transistor is independent of the others. (a) What is the probability that the \(10^{\text {th }}\) transistor produced is the first with a defect? (b) What is the probability that the machine produces no defective transistors in a batch of \(100 ?\) (c) On average, how many transistors would you expect to be produced before the first with a defect? What is the standard deviation? (d) Another machine that also produces transistors has a \(5 \%\) defective rate where each transistor is produced independent of the others. On average how many transistors would you expect to be produced with this machine before the first with a defect? What is the standard deviation? (e) Based on your answers to parts (c) and (d), how does increasing the probability of an event affect the mean and standard deviation of the wait time until success?4.14 Defective rate. A machine that produces a special type of transistor (a component of computers) has a \(2 \%\) defective rate. The production is considered a random process where each transistor is independent of the others. (a) What is the probability that the \(10^{\text {th }}\) transistor produced is the first with a defect? (b) What is the probability that the machine produces no defective transistors in a batch of \(100 ?\) (c) On average, how many transistors would you expect to be produced before the first with a defect? What is the standard deviation? (d) Another machine that also produces transistors has a \(5 \%\) defective rate where each transistor is produced independent of the others. On average how many transistors would you expect to be produced with this machine before the first with a defect? What is the standard deviation? (e) Based on your answers to parts (c) and (d), how does increasing the probability of an event affect the mean and standard deviation of the wait time until success?

Speeding on the \(1-5,\) Part I. The distribution of passenger vehicle speeds traveling on the Interstate 5 Freeway (I-5) in California is nearly normal with a mean of 72.6 miles/hour and a standard deviation of 4.78 miles/hour. (a) What percent of passenger vehicles travel slower than 80 miles/hour? (b) What percent of passenger vehicles travel between 60 and 80 miles/hour? (c) How fast do the fastest \(5 \%\) of passenger vehicles travel? (d) The speed limit on this stretch of the I-5 is 70 miles/hour. Approximate what percentage of the passenger vehicles travel above the speed limit on this stretch of the \(\mathrm{I}-5\).

Area under the curve, Part I. What percent of a standard normal distribution \(N(\mu=0, \sigma=1)\) is found in each region? Be sure to draw a graph. (a) \(Z<-1.35\) (b) \(Z>1.48\) (c) \(-0.42\)

Suppose you're considering buying your expensive chemistry textbook on Ebay. Looking at past auctions suggests that the prices of this textbook follow an approximately normal distribution with mean $$\$ 89$$ and standard deviation $$\$ 15$$. (a) What is the probability that a randomly selected auction for this book closes at more than $$\$ 100 ?$$ (b) Ebay allows you to set your maximum bid price so that if someone outbids you on an auction you can automatically outbid them, up to the maximum bid price you set. If you are only bidding on one auction, what are the advantages and disadvantages of setting a bid price too high or too low? What if you are bidding on multiple auctions? (c) If you watched 10 auctions, roughly what percentile might you use for a maximum bid cutoff to be somewhat sure that you will win one of these ten auctions? Is it possible to find a cutoff point that will ensure that you win an auction? (d) If you are willing to track up to ten auctions closely, about what price might you use as your maximum bid price if you want to be somewhat sure that you will buy one of these ten books?

Bernoulli, the standard deviation. Use the probability rules from Section 3.4 to derive the standard deviation of a Bernoulli random variable, i.e. a random variable \(X\) that takes value 1 with probability \(p\) and value 0 with probability \(1-p .\) That is, compute the square root of the variance of a generic Bernoulli random variable.
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