Sure! Let's approach this step by step to find which values fall within one standard deviation of the mean.
Step 1: Identify the given data and statistical parameters.
- Mean ([tex]\(\mu\)[/tex]): 20.4
- Standard deviation ([tex]\(\sigma\)[/tex]): 7.9
- Data set: 6, 11, 15, 17, 19, 24, 25, 26, 28, 33
Step 2: Calculate the boundaries for one standard deviation from the mean.
- Lower bound = Mean - Standard deviation
[tex]\[ \text{Lower bound} = 20.4 - 7.9 = 12.5 \, (\text{approximately}) \][/tex]
- Upper bound = Mean + Standard deviation
[tex]\[ \text{Upper bound} = 20.4 + 7.9 = 28.3 \, (\text{approximately}) \][/tex]
Step 3: Determine which values fall within the calculated boundaries (12.5 and 28.3).
Let's examine each number in the data set:
- 6: Not within 12.5 and 28.3
- 11: Not within 12.5 and 28.3
- 15: Within 12.5 and 28.3
- 17: Within 12.5 and 28.3
- 19: Within 12.5 and 28.3
- 24: Within 12.5 and 28.3
- 25: Within 12.5 and 28.3
- 26: Within 12.5 and 28.3
- 28: Within 12.5 and 28.3
- 33: Not within 12.5 and 28.3
Step 4: Compile the values that fall within one standard deviation from the mean.
The values that fall within one standard deviation are:
[tex]\[ 15, 17, 19, 24, 25, 26, 28 \][/tex]
Conclusion:
The data values that fall within one standard deviation of the mean are [tex]\( 15, 17, 19, 24, 25, 26, \)[/tex] and [tex]\( 28 \)[/tex].
