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Chapter 6: Problem 9

For the standard normal distribution, what does \(z\) represent?

Short Answer

Expert verified
In the standard normal distribution, 'z' represents the Z-score. The Z-score is a measure that describes a value's relationship to the mean of a group of values. In particular, it's equal to the number of standard deviations a certain data point is away from the mean.

Step by step solution

01

Understand the standard normal distribution

The standard normal distribution, also known as a z-distribution, is a special kind of normal distribution where the mean (\( \mu \)) is zero and the standard deviation (\( \sigma \)) is one.
02

Define the Z-score

In a standard normal distribution, the term 'z' represents the Z-score. A Z-score measures how many standard deviations an element or data point is from the mean.
03

Explain the significance of the Z-score

The Z-score is crucial because it allows us to understand where a particular data point stands compared to the rest of the data set. A Z-score of zero shows that the data point is identical to the mean. If the Z-score is positive, the data point is above average, and if it's negative, the data point is below average.

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Most popular questions from this chapter

Determine the area under a normal distribution curve with \(\mu=55\) and \(\sigma=7\) A. to the right of \(x=58\) b. to the right of \(x=43\) c. to the left of \(x=68\) d. to the left of \(x=22\)

During the 2009 edition of the reality show Britain's Got Talent, runner-up and Internet singing sensation Susan Boyle obtained \(20.2 \%\) of the first- place votes. Suppose that this percentage would hold true for all potential voters (note: the population of interest would be all viewers of ITV, which carries the show). Find the probability that, in a random sample of 250 potential voters, the number who would vote for Susan Boyle is a. exactly 57 b. 35 to 41 c. at least 60

For the standard normal distribution, what is the area within \(2.5\) standard deviations of the mean?

Find the area under the standard normal curve a. from \(z=0\) to \(z=3.94\) \(\mathbf{b}\). between \(z=0\) and \(z=-5.16\) \(\mathrm{c}\). to the right of \(z=5.42\) d. to the left of \(z=-3.68\)

Find the following binomial probabilities using the normal approximation. a. \(n=140, \quad p=.45, \quad P(x=67)\) b. \(n=100, \quad p=.55, \quad P(52 \leq x \leq 60)\) c. \(n=90, \quad p=.42, \quad P(x \geq 40)\) d. \(n=104, p=.75, \quad P(x \leq 72)\)
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