Question

12. Suppose that the mean of a sample of mound-shaped data is 40 and the standard deviation is 4.

(4) a. Use the Empirical rule to state the probability that the data is one, two, and three standard deviations from the mean and state the intervals for each of these.

(4) b. Use the Tchebysheff’s theorem to state the probability that the data is 1, 1.5, 2, and 3 standard deviations from the mean and state the intervals for each of these.

(4) c. What would the x value of the sample be if it was 1.5 standard deviations above the mean?

(4) d. What is the z value if the data point was 50?

(2) e. Using the Empirical rule, what is the probabilty that the data is between 36 and 48? Explain how you made this calculation.

(2) f. Using the standard normal table, what is the probabilty that the data is between 36 and 48? Explain how you made this calculation.

Answer 1

Let X ba random variable having mean
u= 40
and standard deviation $\sigma$ = 4

a. The emprerical rule or the 3 $\sigma$ rule states that for normal distribution 68% of the data falls within 1 standard deviation of mean
tuto)
95% within 2 standard deviation from mean
(n 20
and, 99.7% within 3 standard deviation from mean
(u 30
.

Thus,

P(40 – 4< X < 40 + 4) = 0.68

or, P(36 < X < 44) = 0.68

The required interval is (36,44)

P(40 – 2*4 < X < 40 + 2 * 4) = 0.95

or, P (32 < X < 48) = 0.95

The required interval is (32,48)

.

P(40 – 3*4< X < 40+3 * 4) = 0.997

or, P28 < X<52) = 0.997

The required interval is (28,52)

b. Tchebysheff's theorem states that, for any value of k that is greater than or equals to 1 atleast 100(1-1/k²)% of the data will lie within k standard deviations of the mean. i.e.

P(μ – k και σ<X <μ+k και σ) Σ (1 – 1/2)

For k = 1

P(40 – 1* 4 < X < 40+1 * 4) > (1–1/12)

or, P(36 < X < 44) > 0

The required interval is (36,44)

For, k=1.5

P(40 – 1.5 * 4 < X < 40+ 1.5 * 4) > (1 – 1/1.52)

or, P(34 < X < 46) > 0.56

The required interval is (34,46)

For, k= 2

P(40 – 2* 4 < X < 40+ 2 * 4) > (1 – 1/22)

or, P(32 < X < 48) > 0.75

The required interval is (32,48)

For k=3

P(40 – 3 * 4 < X < 40+ 3 * 4) > (1-1/32)

or, P(28 < X < 52) > 8/9

or, P(28 < X<52) > 0.89

The required interval is (28,52)

(c)  the x value of the sample be if it was 1.5 standard deviations above the mean =

u + 1.5 * 0 = 40 + 1.5*4= 40 + 6 = 46

(d)  the z value if the data point was 50 ,

T 11 o

or, 2 2= 50 – 40 4

= 10/4=2.5