Question

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Part 1
Assume that a normal distribution of data has a mean of 23 and a standard deviation of 5. Use the 68minus95minus99.7 rule to find the percentage of values that lie above 38.
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Part 1
What percentage of values lie above 38​?
  
enter your response here​%​ (Type an integer or a​ decimal.)

Answer 1

To determine the percentage of values that lie above 38 in a normal distribution with a mean of 23 and a standard deviation of 5, we can use the properties of the normal distribution and the 68-95-99.7 rule.

1. **Calculate the Z-score for 38:**
The Z-score formula is:
\[
Z = \frac{(X - \mu)}{\sigma}
\]
where \(X\) is the value, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.

Here, \(X = 38\), \(\mu = 23\), and \(\sigma = 5\).

\[
Z = \frac{(38 - 23)}{5} = \frac{15}{5} = 3
\]

2. **Interpret the Z-score:**
A Z-score of 3 indicates that the value of 38 is 3 standard deviations above the mean.

3. **Apply the 68-95-99.7 rule:**
According to the 68-95-99.7 rule:
- Approximately 68% of the data lies within 1 standard deviation of the mean (between 18 and 28).
- Approximately 95% lies within 2 standard deviations (between 13 and 33).
- Approximately 99.7% lies within 3 standard deviations (between 8 and 38).

Since 99.7% of the data lies within 3 standard deviations (between 8 and 38), that means:
- The remaining percentage outside that range is \(100\% - 99.7\% = 0.3\%\).
- Since the normal distribution is symmetric, this 0.3% is divided equally between the two tails.

4. **Calculate the percentage above 38:**
Therefore, the percentage of values that lie **above** 38 is half of 0.3%:

\[
\frac{0.3\%}{2} = 0.15\%
\]

Thus, the percentage of values that lie above 38 is **0.15%**.