Answer :
Therefore, using the empirical rule, the probability is about 68%.
To estimate the probability that a randomly selected student has a body weight between 116 and 134 pounds using the empirical rule (also known as the 68-95-99.7 rule), follow these steps:
1. Identify the Mean and Standard Deviation:
Mean [tex](\(\mu\))[/tex] = 125 pounds
Standard Deviation [tex](\(\sigma\))[/tex] = 9 pounds
2. Determine the Number of Standard Deviations from the Mean for Each Bound:
For 116 pounds:
[tex]\[ \frac{116 - 125}{9} = \frac{-9}{9} = -1 \][/tex]
So, 116 pounds is 1 standard deviation below the mean.
For 134 pounds:
[tex]\[ \frac{134 - 125}{9} = \frac{9}{9} = 1 \][/tex]
So, 134 pounds is 1 standard deviation above the mean.
3. Apply the Empirical Rule:
The empirical rule states that:
Approximately 68% of the data falls within 1 standard deviation of the mean.
Approximately 95% falls within 2 standard deviations.
Approximately 99.7% falls within 3 standard deviations.
Since 116 and 134 pounds are both within 1 standard deviation of the mean, the probability that a randomly selected student has a body weight between 116 and 134 pounds is approximately 68%.
Complete question is shown below.
A normal distribution is observed from the body weights of the forty students in a class. If the mean is 125 pounds and the standard deviation is 9 pounds, what is the probability that a randomly selected student has a body weight between 116 and 134 pounds? Use the empirical rule.
Using the empirical rule, there's a 68% probability that a randomly selected student has a body weight between 116 and 134 pounds, given a mean of 125 pounds and a standard deviation of 9 pounds.
To answer this question, we will use the empirical rule. The empirical rule states that for a normal distribution:
- 68% of the data falls within one standard deviation of the mean,
- 95% of the data falls within two standard deviations of the mean, and
- 99.7% of the data falls within three standard deviations of the mean.
Given:
Mean (
): 125 pounds
Standard deviation (
x): 9 pounds
The range from 116 to 134 pounds is within one standard deviation of the mean (125 - 9 = 116 and 125 + 9 = 134).
So, according to the empirical rule, the probability that a randomly selected student has a body weight between 116 and 134 pounds is approximately 68%.
