You are given a sample of 25 observations. 7 6 6 11 8 9 11 9 10 8 7 7 5 9 10 7 7

Question

You are given a sample of 25 observations.

7 6 6 11 8 9 11 9 10 8 7 7
5 9 10 7 7 7 7 9 12 10 10 8 6

(a) Compute mean x .

(b) Compute sample variance s2.

(c) Compute sample standard deviation s.

(d) Compute 1–s interval, report how many observations are within 1–s interval, and what percentage

of the total 25 observations are within 1–s interval?

(e) Compute 2–s interval, report how many observations are within 2–s interval, and what percentage

of the total 25 observations are within 2–s interval?

(f) Compute 3–s interval, report how many observations are within 3–s interval, and what percentage

of the total 25 observations are within 3–s interval?

(g) Compare the three percentages obtained in Q.2.d, Q.2.e, and Q.2.f with Chebyshev’s rule (Refer to Lecture Note #8) and tell whether or not these three percentages are consistent with Cheby- shev’s rule.

(h) Compare the three percentages obtained in Q.2.d, Q.2.e, and Q.2.f with the Empirical rule (Refer to Lecture Note #8) and tell whether or not these three percentages are consistent with the Empirical rule.

Explanation / Answer

(a) Mean = 8.24

(b) Sample variance s2 = 3.3567

(c) sample standard deviation s = 1.8321

(d) Compute 1 - s interval = 8.24 +- 1.8321 = (6.41, 10.07)

There are 18 numbers in 1-s interval

percentage of the total 25 observations are within 1–s interval = 18/25 = 0.72

(e) Compute 2 - s interval = 8.24 +- 2 * 1.8321 = (4.58, 11.90)

There are 24 numbers in 1-s interval

percentage of the total 24 observations are within 2–s interval = 24/25 = 0.96

(f)

Compute 3 - s interval = 8.24 +- 3 * 1.8321 = (2.74, 13.74)

There are 25 numbers in 3-s interval

percentage of the total 25 observations are within 3–s interval = 25/25 =1

(g) Cheybhev rue tell that for

2- s interval at least = 75% value shall be there

3-s interval at least = 88.89 % value must be there.

Yes the given values follows Cheybheshev inequality

(h) Empirical rule tells

1-s interval contains 68% of value s

2-s interval contains 95% of value

3-s interval contains 99.7% of value

so the values which we got consistent with empirical rule too.

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