Question

9. The mean value of land and buildings per acre from a sample of farms is ​$1700, with a standard deviation of ​$200. The data set has a​ bell-shaped distribution. Assume the number of farms in the sample is 73.

A. Use the empirical rule to estimate the number of farms whose land and building values per acre are between $1300 and ​$2100.

B. If 21 additional farms were​ sampled, about how many of these additional farms would you expect to have land and building values between $1300 per acre and ​$2100 per​ acre?

10. From a sample with n=28​, the mean number of televisions per household is 2 with a standard deviation of 1 television. Using Chebychev's Theorem, determine at least how many of the households have between 0 and 4 televisions.

Answer 1

Solution

Back-up Theory

Empirical rule, also known as 68 – 95 – 99.7 percent rule: applicable to symmetric (bell-shaped) distributions

P{(µ - σ) ≤ X ≤ (µ + σ)} = 0.68; ..........……………………………………………………………………..………….(1a)

P{(µ - 2σ) ≤ X ≤ (µ + 2σ)} = 0.95; ...............................……………………………………………..……………….(1b)

P{(µ - 3σ) ≤ X ≤ (µ + 3σ)} = 0.997 ......................................………………………………………..……………….(1c)

i.e., Mean ± 1 Standard Deviation holds 68% of the observations; …………………………..…….....………….(1d)

Mean ± 2 Standard Deviations holds 95% of the observations ……………………………………...………..….(1e)

and Mean ± 3 Standard Deviations holds 99.7% of the observations. ……………………………….......……...(1f).

Chebyshev’s Inequality:

If E(X) = µ and V(X) = σ², then P(|X - µ | ≥ kσ) ≤ 1/k² for a wide spectrum of distributions……….............…. (2)

Now to work out the solution,

Q 9

Part (A)

Given the mean value of land and buildings per acre from a sample of farms is ​$1700, with a standard deviation of ​$200,

1300 = 1700 – (2 x 200) and 2100 = 1700 + (2 x 200), i.e., 1300 = (µ - 2σ) and 2100 = (µ+ 2σ).

So, vide (1b), probability that the values per acre are between $1300 and ​$2100 is 0.95.

Since the number of farms in the sample is 73, the estimate of the number of farms whose land and building values per acre are between $1300 and ​$2100 = 73 x 0.95 = 69.35 or

= 69 Answer

Part (B)

By the same principle as above, if 21 additional farms were​ sampled, additional farms expected to have land and building values between $1300 per acre and ​$2100 per​ acre = 21 x 0.95 = 19.95

= 20 Answer 2

Q 10

Let X = number of televisions per household

Given n=28​, the mean number of televisions per household is 2 with a standard deviation of 1,

µ = 2 and σ = 1, 0 = µ - 2σ and 4 = µ + 2σ.

So, P[0 < X < 4]

= P[|x - µ |< 2σ]

Vide (2), k = 2 or 1/k² = ¼

P[|x - µ |< 2σ] = 1 - P[|x - µ |≥ 2σ]

So, by Chebyshev’s Inequality,

P[0 < X < 4] = 1 - P[|x - µ |≥ 2σ] ≥ 1 – ¼ = 0.75.

Thus, probability that the households have between 0 and 4 televisions

= P[0 < X < 4] ≥ 0.75

So, out of 28 houses, at least (28 X 0.75) = 21

So, at least 21 households will have between 0 and 4 televisions Answer

DONE