Chapter 8: Q 8.109. (page 342)

A variable has a mean of $100$ and a standard deviation of $16$ Four observations of this variable have a mean of $108$ and a sample standard deviation of $12$ . Determine the observed value of the
a. standardized version of $\bar{x}$ .
b. studentized version of $\bar{x}$ .

Short Answer

Expert verified

Part (a) $1$

Part (b) $1.33$

Step by step solution

Part (a) Step 1: Given information

Sample size $n = 4$

Sample mean $\bar{x} = 108$

Sample s.d., $s = 12$

Part (a) Step 2: Concept

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

Part (a) Step 3: Calculation

Let the variable $x$ has meant, $μ = 100$ and s.d., $σ = 16$

A standardized version of $\bar{x}$ ,

$Z = \frac{\bar{x} - μ}{\frac{σ}{\sqrt{n}}} = \frac{108 - 100}{\frac{16}{\sqrt{4}}} = \frac{8}{16 / 2} = \frac{8}{8} = 1$

Part (b) Step 1: Calculation

Studentized version of $\bar{x}$

$t = \frac{\bar{x} - μ}{\frac{s}{\sqrt{n}}} = \frac{108 - 100}{\frac{12}{\sqrt{4}}} = \frac{8}{12 / 2} = \frac{8}{6} = 1.33$

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A variable has a mean of $100$ and a standard deviation of $16$ Four observations of this variable have a mean of $108$ and a sample standard deviation of $12$ . Determine the observed value of the
a. standardized version of $\bar{x}$ .
b. studentized version of $\bar{x}$ .

Short Answer

Step by step solution

Part (a) Step 1: Given information

Part (a) Step 2: Concept

Part (a) Step 3: Calculation

Part (b) Step 1: Calculation

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A variable has a mean of 100 and a standard deviation of 16 Four observations of this variable have a mean of 108 and a sample standard deviation of 12 . Determine the observed value of thea. standardized version of x¯.b. studentized version of x¯.

Short Answer

Step by step solution

Part (a) Step 1: Given information

Part (a) Step 2: Concept

Part (a) Step 3: Calculation

Part (b) Step 1: Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Calculus

Geometry

Theoretical and Mathematical Physics

Statistics

Logic and Functions

Study anywhere. Anytime. Across all devices.

A variable has a mean of $100$ and a standard deviation of $16$ Four observations of this variable have a mean of $108$ and a sample standard deviation of $12$ . Determine the observed value of the
a. standardized version of $\bar{x}$ .
b. studentized version of $\bar{x}$ .