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To solve this question, we can use the [tex]$68-95-99.7$[/tex] rule, also known as the empirical rule, which tells us about the spread of data in a normal distribution.
First, let's restate the key values in our problem:
- The mean ([tex]\(\mu\)[/tex]) of the distribution is 18.
- The standard deviation ([tex]\(\sigma\)[/tex]) is 3.
- We want to find the percentage of values that lie below 24.
### Step 1: Determine How Many Standard Deviations Away 24 Is From the Mean
We need to calculate how many standard deviations away the value 24 is from the mean.
[tex]\[ z = \frac{(X - \mu)}{\sigma} = \frac{(24 - 18)}{3} = 2 \][/tex]
So, 24 is 2 standard deviations above the mean.
### Step 2: Apply the [tex]$68-95-99.7$[/tex] Rule
The [tex]$68-95-99.7$[/tex] rule tells us:
- Approximately 68% of the data falls within 1 standard deviation of the mean.
- Approximately 95% of the data falls within 2 standard deviations of the mean.
- Approximately 99.7% of the data falls within 3 standard deviations of the mean.
Since 24 is 2 standard deviations above the mean, we use the rule that states:
- About 95% of the data lies within 2 standard deviations of the mean.
### Step 3: Calculate the Percentage Below 24
Considering the normal distribution's symmetry:
- Half of the data (50%) lies below the mean.
- The other half (50%) lies above the mean.
Since 24 is 2 standard deviations above the mean, this means we are looking at the cumulative percentage up to this point:
- 95% of the data lies within 2 standard deviations, which means from the lowest extreme up to [tex]\(18 + 2 \times 3 = 24\)[/tex].
To determine the percentage of values that lie below 24:
- We subtract the 2.5% above the 2 standard deviations:
[tex]\[ 100\% - 50\% + 34.1\% = 84.1\% \][/tex]
### Conclusion
Therefore, the percentage of values that lie below 24 in this normal distribution is 84.1%.
First, let's restate the key values in our problem:
- The mean ([tex]\(\mu\)[/tex]) of the distribution is 18.
- The standard deviation ([tex]\(\sigma\)[/tex]) is 3.
- We want to find the percentage of values that lie below 24.
### Step 1: Determine How Many Standard Deviations Away 24 Is From the Mean
We need to calculate how many standard deviations away the value 24 is from the mean.
[tex]\[ z = \frac{(X - \mu)}{\sigma} = \frac{(24 - 18)}{3} = 2 \][/tex]
So, 24 is 2 standard deviations above the mean.
### Step 2: Apply the [tex]$68-95-99.7$[/tex] Rule
The [tex]$68-95-99.7$[/tex] rule tells us:
- Approximately 68% of the data falls within 1 standard deviation of the mean.
- Approximately 95% of the data falls within 2 standard deviations of the mean.
- Approximately 99.7% of the data falls within 3 standard deviations of the mean.
Since 24 is 2 standard deviations above the mean, we use the rule that states:
- About 95% of the data lies within 2 standard deviations of the mean.
### Step 3: Calculate the Percentage Below 24
Considering the normal distribution's symmetry:
- Half of the data (50%) lies below the mean.
- The other half (50%) lies above the mean.
Since 24 is 2 standard deviations above the mean, this means we are looking at the cumulative percentage up to this point:
- 95% of the data lies within 2 standard deviations, which means from the lowest extreme up to [tex]\(18 + 2 \times 3 = 24\)[/tex].
To determine the percentage of values that lie below 24:
- We subtract the 2.5% above the 2 standard deviations:
[tex]\[ 100\% - 50\% + 34.1\% = 84.1\% \][/tex]
### Conclusion
Therefore, the percentage of values that lie below 24 in this normal distribution is 84.1%.
Step-by-step explanation:
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