Question

16 Use the Empirical Rule. (Do not use technology. Technology will give a slightly different answer.)

Draw a sketch of the normal distribution and label the mean and 1, 2, and 3 standard devitons above and
below the mean with computed values.
Assume that the weight of 1-year-old girls in the USA is normally distributed with Mean = 9.5 kg, Standard
Deviation = 1.1 kg.
(a) 68% of the data is between which 2 values? 8.4 oro kg -- 10.0 kg
(b) 95% of the data is between which 2 values? 7.3 kg - 11. oo kg
(c) What percentage of the data is less than 8.4 kg?
(d) What percentage of the data is between 7.3 kg and 11.7 kg?
(w) What percentage of the data is more than 12.8 kg?

Answer 1

a)8.4kg to 10.6kg

b)7.3 kg to 11.7kg

c)16%

d) 95%

e) 0.15%

Explanation:

The empirical rule formula states that:

68% of data falls within 1 standard deviation from the mean - that means between μ - σ and μ + σ .

95% of data falls within 2 standard deviations from the mean - between μ – 2σ and μ + 2σ .

99.7% of data falls within 3 standard deviations from the mean - between μ - 3σ and μ + 3σ

From the question,

Mean(μ) = 9.5 kg,

Standard Deviation (σ) = 1.1 kg.

a) 68% of the data is between which 2 values?

68% of data falls within 1 standard deviation from the mean - that means between μ - σ and μ + σ .

μ - σ

= 9.5kg - 1.1kg

= 8.4kg

μ + σ

= 9.5kg + 1.1kg

= 10.6kg

Therefore, 68% of the data is between 8.4kg to 10.6kg

(b) 95% of the data is between which 2 values?

95% of data falls within 2 standard deviations from the mean - between μ – 2σ and μ + 2σ .

μ - 2σ

9.5kg - 2(1.1)kg

9.5kg - 2.2 kg

= 7.3kg

μ + 2σ

9.5kg + 2(1.1)kg

9.5kg + 2.2 kg

= 11.7kg

Therefore, 95% of the data is between 7.3 kg to 11.7kg

(c) What percentage of the data is less than 8.4 kg?

We need to find how many standard deviations from the mean 8.4kg is, in order to know the percentage it falls under

μ - xσ = 8.4kg

= 9.5kg - 1.1x = 8.4kg

9.5 - 8.4 = 1.1x

1.1 = 1.1x

x = 1.1/1.1

x = 1

Hence, the percentage it falls under according to Empirical rule = 68%

The percentage of the data that is less than 8.4 kg is calculated as:

100 - 68/2 = 32/2 = 16%

(d) What percentage of the data is between 7.3 kg and 11.7 kg?

We have to find the Standard deviation that they fall under

For 7.3 kg

μ - xσ = 7.3kg

= 9.5kg - 1.1x = 7.3kg

9.5 - 7.3 = 1.1x

2.2 = 1.1x

x = 2.2/1.1

x = 2

μ + xσ = 11.7kg

= 9.5kg + 1.1x = 11.7kg

11.7 - 9.5 = 1.1x

2.2 = 1.1x

x = 2.2/1.1

x = 2

Therefore, based on the calculation above and the empirical formula rule, the percentage of the data is between 7.3 kg and 11.7 kg is 95%

(e) What percentage of the data is more than 12.8 kg?

μ + xσ = 12.8kg

9.5kg + 1.1x = 12.8kg

= 12.8- 9.5 = 1.1x

3.3 = 1.1x

3.3 = 1.1x

x = 3.3/1.1

x = 3

Hence, the percentage it falls under according to Empirical rule = 99.7%

The percentage of the data that is more than 12.8kg is calculated as:

100 - 99.7/2 = 0.3/2 = 0.15%