Chapter 3: Q. 3.118 (page 122)

A quantitative data set of size $200$ has mean $20$ and standard deviation $4$ . At least how many observations lie between $8$ and $32$ $?$

Short Answer

Expert verified

At least $178$ observations lie between $8$ and $32$ .

Step by step solution

Given information.

We have been given that,

Mean= $20$ , Standard deviation = $4$ , data set= $200$

We need find the number of observations lie between $8$ and $32$ .

Simplify.

Chebyshev's rule: At least $100 (1 - \frac{1}{k^{2}}) %$ of the data values lie within $k$ standard deviation from the mean $(k > 1)$ .

Using Chebyshev's Rule with $k = 3$ , we know that at least

$100 (1 - \frac{1}{k^{2}}) % = 100 (1 - \frac{1}{3^{2}}) % = 100 (\frac{8}{9}) % = 89 %$

is within three standard deviation from the mean.

Determine the value that are three standard deviation from the mean,

$20 - 3 \times 4 = 8 20 + 3 \times 4 = 32$

At least $89 %$ of the observations lie between $8$ and $32$ .

The number of observation is the product of the percentage and sample size

$89 % \times 200 = 178$

Thus , at least $178$ observations lie between $8$ and $32$ .

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A quantitative data set of size $200$ has mean $20$ and standard deviation $4$ . At least how many observations lie between $8$ and $32$ $?$

Short Answer

Step by step solution

Given information.

Simplify.

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A quantitative data set of size 200has mean 20and standard deviation 4. At least how many observations lie between8and32?

Short Answer

Step by step solution

Given information.

Simplify.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Applied Mathematics

Decision Maths

Mechanics Maths

Logic and Functions

Statistics

Study anywhere. Anytime. Across all devices.

A quantitative data set of size $200$ has mean $20$ and standard deviation $4$ . At least how many observations lie between $8$ and $32$ $?$