Chapter 5: Q55SE (page 295)

:A random sample of n = 68 observations is selected from a population with $μ = 19.6$ and $σ = 3.2$ Approximate each of the following probabilities
a) $p (\bar{X} ⩽ 19.6)$
b) $p (\bar{X} ⩽ 19)$
c) $p (\bar{X} ⩾ 20.1)$
d) $p (19.2 ⩽ \bar{X} ⩽ 20.6)$

Short Answer

Expert verified

Step by step solution

Given Information

The number of sample size is 68.

The mean and standard deviation are 19.6 and 3.2

(a) Compute the probability for given condition

$(\bar{X} ⩽ 19.6) = p (\frac{\bar{x} - μ}{\frac{σ}{\sqrt{n}}} ⩽ \frac{19.6 - 19.6}{\frac{3.2}{\sqrt{68}}}) = p (z ⩽ \frac{0}{0.388}) = 0.5$

Therefore, $p (\bar{X} ⩽ 19.6) = 0.5$

(b) Calculation

$(\bar{X} ⩽ 19) = p (\frac{\bar{x} - μ}{\frac{σ}{\sqrt{n}}} ⩽ \frac{19 - 19.6}{\frac{3.2}{\sqrt{68}}}) = p (z ⩽ \frac{- 0.6}{0.388}) = 0.061$

Therefore $p (\bar{X} ⩽ 19) = 0.061$

(c) Calculation

$(\bar{X} ⩾ 20.1) = p (\frac{\bar{x} - μ}{\frac{σ}{\sqrt{n}}} ⩽ \frac{20.1 - 19.6}{\frac{3.2}{\sqrt{68}}}) = p (z ⩽ \frac{0.5}{0.388}) = 0.098$

$p (\bar{X} ⩽ 19) = 0.061$

(d) Calculation

$p (19.2 ⩽ \bar{X} ⩽ 20.6) = p (\frac{19.2 - 19.6}{\frac{3.2}{\sqrt{68}}} ⩽ \frac{\bar{x} - μ}{\frac{σ}{\sqrt{n}}} ⩽ \frac{20.6 - 19.6}{\frac{3.2}{\sqrt{68}}}) = p (\frac{- 0.4}{0.388} ⩽ z ⩽ \frac{1}{0.388}) = p (z ⩽ \frac{1}{0.388}) - p (z ⩽ \frac{- 0.4}{0.388}) = 0.99 - 0.15 = 0.84$

Therefore, $p (19.2 ⩽ \bar{X} ⩽ 20.6) = 0.84$

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:A random sample of n = 68 observations is selected from a population with $μ = 19.6$ and $σ = 3.2$ Approximate each of the following probabilities
a) $p (\bar{X} ⩽ 19.6)$
b) $p (\bar{X} ⩽ 19)$
c) $p (\bar{X} ⩾ 20.1)$
d) $p (19.2 ⩽ \bar{X} ⩽ 20.6)$

Short Answer

Step by step solution

Given Information

(a) Compute the probability for given condition

(b) Calculation

(c) Calculation

(d) Calculation

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:A random sample of n = 68 observations is selected from a population withμ=19.6and σ=3.2Approximate each of the following probabilitiesa)pX¯⩽19.6b)pX¯⩽19c)pX¯⩾20.1d)p19.2⩽X¯⩽20.6

Short Answer

Step by step solution

Given Information

(a) Compute the probability for given condition

(b) Calculation

(c) Calculation

(d) Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Applied Mathematics

Statistics

Decision Maths

Geometry

Calculus

Study anywhere. Anytime. Across all devices.

:A random sample of n = 68 observations is selected from a population with $μ = 19.6$ and $σ = 3.2$ Approximate each of the following probabilities
a) $p (\bar{X} ⩽ 19.6)$
b) $p (\bar{X} ⩽ 19)$
c) $p (\bar{X} ⩾ 20.1)$
d) $p (19.2 ⩽ \bar{X} ⩽ 20.6)$