Chapter 6: Q.6.113 (page 279)

Let $0 < α < 1$ . Show that for a regularly distributed variable $100 (1 - α) %$ of all possible observations lie within $z_{α / 2}$ standard deviations to either side of the mean, that is, between $μ - z_{F / 2} \cdot σ$ and $μ + z_{σ / 2} \cdot σ$ .

Short Answer

Expert verified

Proved that $100 (1 - α) %$ of all possible observations lie between $μ - z_{α / 2} \cdot σ$ and $μ + z_{α / 2} \cdot σ$ .

Step by step solution

Given Information

Normal distribution, $0 < α < 1$ .

Explanation

$a$ = the possibility. In a confidence interval, the population parameter will be left out.; $1 - a$ = the probability of the population parameter being included in the interval. With probability, the true value of a population parameter. $1 - a$ will be included in the $100 (1 - a)$ percent confidence range.

Consider $p (| z | \leq z_{α / 2}) = 1 - α$

$p (- z_{α / 2} \leq z \leq z_{α / 2}) = 1 - α$

$p (- z_{α / 2} \leq \frac{x - μ}{σ} \leq z_{α / 2}) = 1 - α (∵ z = \frac{x - μ}{σ})$

$p (μ - z_{α / 2} \cdot σ \leq x \leq μ + z_{α / 2} \cdot σ) = 1 - α$

So, that $100 (1 - α) %$ of all possible observations lie between $μ - z_{α / 2} \cdot σ$ and $μ + z_{α / 2} \cdot σ$ .

Hence, proved.

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Let $0 < α < 1$ . Show that for a regularly distributed variable $100 (1 - α) %$ of all possible observations lie within $z_{α / 2}$ standard deviations to either side of the mean, that is, between $μ - z_{F / 2} \cdot σ$ and $μ + z_{σ / 2} \cdot σ$ .

Short Answer

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Given Information

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Let 0<α<1. Show that for a regularly distributed variable 100(1-α)%of all possible observations lie within zα/2standard deviations to either side of the mean, that is, between μ-zF/2·σand μ+zσ/2·σ.

Short Answer

Step by step solution

Given Information

Explanation

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Most popular questions from this chapter

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Study anywhere. Anytime. Across all devices.

Let $0 < α < 1$ . Show that for a regularly distributed variable $100 (1 - α) %$ of all possible observations lie within $z_{α / 2}$ standard deviations to either side of the mean, that is, between $μ - z_{F / 2} \cdot σ$ and $μ + z_{σ / 2} \cdot σ$ .