Question

Suppose the data have a bell-shaped distribution with a mean of 25 and a standard deviation of 5. Use the empirical rule to determine the percentage of data within each of the following ranges.
(a) 15 to 35 approximately 68% approximately 95% almost all
(b) 10 to 40 approximately 68% approximately 95% almost all
(c) 20 to 30 approximately 68% approximately 95% almost all

Answer 1

Answer:

a) 15 to 35 approximately 95%

(b) 10 to 40 approximately almost all

(c) 20 to 30 approximately 68%

Explanation:

The data have a bell-shaped distribution which means the data is equally distributed on both sides of the mean.

We have the mean at 25 and a standard deviation of 5 which means that the interval is for each of the values of 5 .

The mean would be u and

The first value would be u ±σ = 25 ± 5= 20 and 30 (68 % )

The second value will be u ± 2σ= 25± 10 = 15 and 35 (95%)

The third value will be u ± 3σ= 25 ± 15 = 10 and 40 (99.7 % almost all)

In the figure below the light blue region gives u ±σ on both sides of the mean

, dark blue gives u ± 2σ values on both sides of the mean and grey gives

u ± 3 σ values on both sides of the mean.

It is obvious that 68 % of the data is contained in the u ±σ light blue region, 95 % of the data in the  u ± 2σ dark blue including light blue and 99.7 % in the u ± 3σ all colored regions.