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Chapter 6: Problem 5

Briefly describe the standard normal distribution curve.

Short Answer

Expert verified
The standard normal distribution, also known as the z-distribution, is a special type of normal distribution where the mean is 0 and the standard deviation is 1. It is symmetrical and bell-shaped, like any normal distribution, but is standard in that any value on the curve can be read in terms of standard deviations above or below the mean.

Step by step solution

01

Understand Normal Distribution

The normal distribution is a probability function that describes how the values of a variable are distributed. It is a symmetric distribution where most of the observations cluster around the central peak and the probabilities for values further away from the mean taper off equally in both directions. Extreme values in both tails of the distribution are similarly unlikely.
02

Define the Standard Normal Distribution

The standard normal distribution is a specific type of normal distribution. It is symmetric like any normal distribution and has a peak at 0, but what differentiates it is that the unit on the X-axis is expressed in terms of standard deviations away from the mean. Therefore, the mean, mode and median of a standard normal distribution is always 0.
03

Explaining Standardization

Standardization is mathematical conversion of a normally distributed variable to a standard normal distribution. Each value gets transformed into a Z-score, which represents the distance between that original value and the mean, expressed in standard deviations. This standardization process allows us to compare values even if the original variables were measured in different units.

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Most popular questions from this chapter

For a binomial probability distribution, \(n=80\) and \(p=.50\). Let \(x\) be the number of successes in 80 trials. a. Find the mean and standard deviation of this binomial distribution. b. Find \(P(x \geq 42)\) using the normal approximation. c. Find \(P(41 \leq x \leq 48)\) using the normal approximation.

Find the following probabilities for the standard normal distribution. a. \(P(z<-1.31)\) b. \(P(1.23 \leq z \leq 2.89)\) c. \(P(-2.24 \leq z \leq-1.19)\) d. \(P(z<2.02)\)

Tommy Wait, a minor league baseball pitcher, is notorious for taking an excessive amount of time between pitches. In fact, his times between pitches are normally distributed with a mean of 36 seconds and a standard deviation of \(2.5\) seconds. What percentage of his times between pitches are a. longer than 39 seconds? b. between 29 and 34 seconds?

Find the following probabilities for the standard normal distribution. a. \(P(z<-1.31)\) b. \(P(1.23 \leq z \leq 2.89)\) c. \(P(-2.24 \leq z \leq-1.19)\) d. \(P(z<2.02)\)

Do the width and/or height of a normal distribution change when its standard deviation remains the same but its mean increases?
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