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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

Stenographer's typos. A very skilled court stenographer makes one typographical error (typo) per hour on average. (a) What probability distribution is most appropriate for calculating the probability of a given number of typos this stenographer makes in an hour? (b) What are the mean and the standard deviation of the number of typos this stenographer makes? (c) Would it be considered unusual if this stenographer made 4 typos in a given hour? (d) Calculate the probability that this stenographer makes at most 2 typos in a given hour.

Playing darts. Calculate the following probabilities and indicate which probability distribution model is appropriate in each case. A very good darts player can hit the bull's eye (red circle in the center of the dart board) \(65 \%\) of the time. What is the probability that he (a) hits the bullseye for the \(10^{\text {th }}\) time on the \(15^{\text {th }}\) try? (b) hits the bullseye 10 times in 15 tries? (c) hits the first bullseye on the third try?

Triathlon times, Part II. In Exercise 4.4 we saw two distributions for triathlon times: \(N(\mu=4313, \sigma=\) 583 ) for Men, Ages 30 - 34 and \(N(\mu=5261, \sigma=807)\) for the Women, Ages \(25-29\) group. Times are listed in seconds. Use this information to compute each of the following: (a) The cutoff time for the fastest \(5 \%\) of athletes in the men's group, i.e. those who took the shortest \(5 \%\) of time to finish. (b) The cutoff time for the slowest \(10 \%\) of athletes in the women's group.

Sickle cell anemia. Sickle cell anemia is a genetic blood disorder where red blood cells lose their flexibility and assume an abnormal, rigid, "sickle" shape, which results in a risk of various complications. If both parents are carriers of the disease, then a child has a \(25 \%\) chance of having the disease, \(50 \%\) chance of being a carrier, and \(25 \%\) chance of neither having the disease nor being a carrier. If two parents who are carriers of the disease have 3 children, what is the probability that (a) two will have the disease? (b) none will have the disease? (c) at least one will neither have the disease nor be a carrier? (d) the first child with the disease will the be \(3^{r d}\) child?

CAPM. The Capital Asset Pricing Model (CAPM) is a financial model that assumes returns on a portfolio are normally distributed. Suppose a portfolio has an average annual return of \(14.7 \%\) (i.e. an average gain of \(14.7 \%\) ) with a standard deviation of \(33 \%\). A return of \(0 \%\) means the value of the portfolio doesn't change, a negative return means that the portfolio loses money, and a positive return means that the portfolio gains money. (a) What percent of years does this portfolio lose money, i.e. have a return less than \(0 \% ?\) (b) What is the cutoff for the highest \(15 \%\) of annual returns with this portfolio?
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