Question

use the empirical rule. draw a sketch of the normal distribution and label the mean and 1, 2, and 3 standard deviations 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? kg -- kg (b) 95% of the data is between which 2 values? kg -- 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?

Explanation:

Step1: Recall the Empirical Rule for 68%

The Empirical Rule states that about 68% of the data in a normal - distribution lies within 1 standard deviation of the mean. The formula for the lower bound is $\mu-\sigma$ and the upper bound is $\mu + \sigma$, where $\mu$ is the mean and $\sigma$ is the standard deviation. Given $\mu = 9.5$ kg and $\sigma=1.1$ kg.
Lower bound: $9.5 - 1.1=8.4$ kg
Upper bound: $9.5 + 1.1 = 10.6$ kg

Step2: Recall the Empirical Rule for 95%

About 95% of the data in a normal - distribution lies within 2 standard deviations of the mean. The formula for the lower bound is $\mu - 2\sigma$ and the upper bound is $\mu+2\sigma$.
Lower bound: $9.5-2\times1.1=9.5 - 2.2 = 7.3$ kg
Upper bound: $9.5 + 2\times1.1=9.5+2.2 = 11.7$ kg

Step3: Analyze the value 8.4 kg

Since 8.4 kg is 1 standard deviation below the mean ($\mu-\sigma$), and the area to the left of $\mu-\sigma$ is $\frac{100 - 68}{2}=16\%$ according to the Empirical Rule.

Step4: Analyze the values 7.3 kg and 11.7 kg

Since 7.3 kg is $\mu - 2\sigma$ and 11.7 kg is $\mu+2\sigma$, about 95% of the data lies between these two values according to the Empirical Rule.

Step5: Analyze the value 12.8 kg

Since $12.8=9.5 + 3\times1.1$ (3 standard deviations above the mean), the area to the right of $\mu + 3\sigma$ is $\frac{100 - 99.7}{2}=0.15\%$ according to the Empirical Rule.

Answer:

(a) 8.4 kg -- 10.6 kg
(b) 7.3 kg -- 11.7 kg
(c) 16%
(d) 95%
(w) 0.15%