Chapter 8: Q 8.113. (page 342)

Explain why there is more variation in the possible values of the studentized version of $\bar{x}$ than in the possible values of the standardized version of $\vec{x}$

Short Answer

Expert verified

Variation in $\bar{x}$ is the single cause of variation in $Z$ values, but variation in $t$ values is produced by both $\bar{x}$ and $s$ The variety in $t$ values is obviously bigger than the variation in $Z$ values.

Step by step solution

Given information

The standardized version of $\bar{x}$ is given by, $Z = \frac{\bar{x} - μ}{\frac{σ}{\sqrt{n}}}$

Concept

The formula used: $Z = \frac{\bar{x} - μ}{\frac{σ}{\sqrt{n}}}$ , $t = \frac{\bar{x} - μ}{\frac{s}{\sqrt{n}}}$

Calculation

Standardized version of $\bar{x}$ is given by,

$Z = \frac{\bar{x} - μ}{\frac{σ}{\sqrt{n}}}$ , Where $σ$ is known. And studentized version of $\bar{x}$ is given by,

$t = \frac{\bar{x} - μ}{\frac{s}{\sqrt{n}}}$ , Where $s$ is the sample S.D $s = \sqrt{\frac{1}{n - 1} \sum_{i = 1}^{n} {(x_{i} - \bar{x})}^{2}}$

Explanation

We can see from the preceding two formulas that the standardized version of $\bar{x}, Z$ , is dependent on only one random variable, $\bar{x}$ The studentized form of $\bar{x}$ , on the other hand, is dependent on two random variables, $\bar{x}$ and $s$ Thus, variation in the values of $Z$ is caused solely by variation in $\bar{x}$ , but variation in the values of $t$ is caused by both $\bar{x}$ and $s$ . Obviously, the variation in $t$ is greater than the variation in $Z$ values.

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Explain why there is more variation in the possible values of the studentized version of $\bar{x}$ than in the possible values of the standardized version of $\vec{x}$

Short Answer

Step by step solution

Given information

Concept

Calculation

Explanation

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Explain why there is more variation in the possible values of the studentized version of x¯ than in the possible values of the standardized version of x→

Short Answer

Step by step solution

Given information

Concept

Calculation

Explanation

One App. One Place for Learning.

Most popular questions from this chapter

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Explain why there is more variation in the possible values of the studentized version of $\bar{x}$ than in the possible values of the standardized version of $\vec{x}$