Certainly! Let's tackle each part step by step using the Empirical Rule, which states that in a normal distribution:
- Approximately 68% of data values fall within one standard deviation of the mean.
- Approximately 95% of data values fall within two standard deviations of the mean.
- Approximately 99.7% of data values fall within three standard deviations of the mean.
Given:
- Number of students: 1100
- Mean score: 82
- Standard deviation: 6
### How many students scored between 76 and 88?
The range between 76 and 88 is within one standard deviation of the mean (82 ± 6). According to the Empirical Rule, approximately 68% of scores fall within one standard deviation of the mean.
Hence, the number of students scoring between 76 and 88 is:
[tex]\[ 0.6826 \times 1100 = 750.86 \][/tex]
### How many students scored between 70 and 94?
The range between 70 and 94 is within two standard deviations of the mean (82 ± 2*6). According to the Empirical Rule, approximately 95% of scores fall within two standard deviations of the mean.
Hence, the number of students scoring between 70 and 94 is:
[tex]\[ 0.9544 \times 1100 = 1049.84 \][/tex]
### How many students scored between 82 and 88?
The range between 82 and 88 can be seen as half of one standard deviation above the mean (82 to 82 + 6). In a normal distribution, this corresponds to approximately half (34%) of the 68% interval above the mean.
Hence, the number of students scoring between 82 and 88 is:
[tex]\[ 0.3085 \times 1100 = 339.35 \][/tex]
### How many students scored lower than 76?
The score of 76 is one standard deviation below the mean. According to the Empirical Rule, approximately (100% - 68%)/2 = 16% of scores fall below one standard deviation below the mean.
Hence, the number of students scoring lower than 76 is:
[tex]\[ 0.1587 \times 1100 = 174.57 \][/tex]
### How many students scored lower than 88?
The score of 88 is one standard deviation above the mean. According to the Empirical Rule, approximately (68% + 0.16)=68% + 16% of data values fall below one standard deviation above the mean.
Hence, the number of students scoring lower than 88 is:
[tex]\[ 0.8413 \times 1100 = 925.43 \][/tex]
In summary:
- Students scoring between 76 and 88: 750.86
- Students scoring between 70 and 94: 1049.84
- Students scoring between 82 and 88: 339.35
- Students scoring lower than 76: 174.57
- Students scoring lower than 88: 925.43
