Let's solve the problem step-by-step and find the percentage of data within the specified ranges for a data set with a mean ([tex]$\mu$[/tex]) of 275 and a standard deviation ([tex]$\sigma$[/tex]) of 25:
1. Calculating the percentage of data between 250 and 300:
To find this percentage, we need to determine the area under the normal distribution curve between these two points.
[tex]\[ \text{Percent between 250 and 300} \approx 68.27\% \][/tex]
2. Calculating the percentage of data between 300 and 350:
Similarly, the area under the normal distribution curve between 300 and 350 needs to be calculated.
[tex]\[ \text{Percent between 300 and 350} \approx 15.73\% \][/tex]
3. Calculating the percentage of data between 200 and 350:
This involves finding the area under the normal distribution curve from 200 to 350.
[tex]\[ \text{Percent between 200 and 350} \approx 99.73\% \][/tex]
4. Calculating the percentage of data between 225 and 300:
The area under the normal distribution curve from 225 to 300 needs to be determined.
[tex]\[ \text{Percent between 225 and 300} \approx 81.86\% \][/tex]
5. Calculating the percentage of data below 275:
The percentage of the data below the mean of 275.
[tex]\[ \text{Percent below 275} = 50\% \][/tex]
6. Calculating the percentage of data between 275 and 325:
Lastly, the area under the normal distribution curve from 275 to 325.
[tex]\[ \text{Percent between 275 and 325} \approx 47.72\% \][/tex]
To summarize:
- The percent of data between 250 and 300 is approximately 68.27%.
- The percent of data between 300 and 350 is approximately 15.73%.
- The percent of data between 200 and 350 is approximately 99.73%.
- The percent of data between 225 and 300 is approximately 81.86%.
- The percent of data below 275 is 50%.
- The percent of data between 275 and 325 is approximately 47.72%.
