Chapter 5: Q20E (page 314)

A random sample of n=900 observations is selected from a population with $μ = 100 and σ = 10$
a. What are the largest and smallest values of $\bar{x}$ that you would expect to see?
b. How far, at the most, would you expect $x$ to deviate from $μ$ ?
c. Did you have to know $μ$ to answer part b? Explain.

Short Answer

Expert verified

$a . The smallest value of x is 99, and the largest value of x is 101 . b . It would not be expected that x deviates from μ c . No, because previous answer only dependent on the standard deviation of the sampling distribution of the sample mean, but not the mean itself .$

Step by step solution

Given information

$A random sample of n = 900 observations is selected from a population with μ = 100 and σ = 10$

Finding the largest and smallest values of x

Here, the sample mean is $μ_{x} = μ$ and the sample standard deviation is $σ_{x} = \frac{σ}{\sqrt{\sqrt{n}}}$

Therefore,

$m_{x} = 100,$

$σ_{x} = \frac{10}{\sqrt{\sqrt{900}}} = \frac{10}{30} = 0.33$

Since, almost all of the time, the sample mean will be within three standard deviations of the mean.

So,

$(μ \pm 3 σ) = (100 \pm 3 (0.33)) = (100 - 0.99, 100 + 0.99) \approx (99, 101)$

Therefore, the smallest value of $x$ is 99, and the largest value of $x$ is 101.

Checking whether deviation of X from μ expect or not.

No, because if more than three standard deviation then

$3 σ_{x} = 3 (\frac{1}{3}) = 1$

Therefore, It would not be expected that that $x$ deviates from $μ$ .

Checking whether μ is necessary to answer part (b) or not.

No, because previous answer only dependent on the standard deviation of the sampling distribution of the sample mean, but not the mean itself.

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Short Answer

Step by step solution

Given information

Finding the largest and smallest values of x

Checking whether deviation of X from μ expect or not.

Checking whether μ is necessary to answer part (b) or not.

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A random sample of n=900 observations is selected from a population with μ=100andσ=10a. What are the largest and smallest values ofx¯ that you would expect to see?b. How far, at the most, would you expect xto deviate from μ?c. Did you have to know μto answer part b? Explain.

Short Answer

Step by step solution

Given information

Finding the largest and smallest values of x

Checking whether deviation of X from μ expect or not.

Checking whether μ is necessary to answer part (b) or not.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Logic and Functions

Mechanics Maths

Statistics

Calculus

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.