The heights of 1000 students in a college are normally distributed with a mean 5’10” and SD 2”. Find the approximate number of students in each range of the heights:
1) 5’8”-6’
2) 5’6”-6’2”
3) Above 5’10”
4) Below 6’
5) Above 5’8”
6) 5’8”-6’4”
*Use 68% for 1 SD, 94% for 2 SD and 98% for 3 SD. (Hint: draw a normal curve, divide the curve into 1, 2 and 3 SD and then put 1%, 2%, 13%, 34%, 34%, 13%, 2%, and 1% in each section from left to right.)
34%
To find the approximate number of students in each range of heights, we can use the normal distribution and the given percentages for each standard deviation (SD). Here's how to calculate the approximate number of students in each range:
1) 5'8" - 6':
- This range corresponds to within 1 SD from the mean.
- According to the normal distribution, 68% of the data falls within 1 SD from the mean.
- Therefore, approximately 68% of the 1000 students will fall within this range.
2) 5'6" - 6'2":
- This range corresponds to within 2 SDs from the mean.
- According to the normal distribution, 94% of the data falls within 2 SDs from the mean.
- Therefore, approximately 94% of the 1000 students will fall within this range.
3) Above 5'10":
- This range corresponds to above the mean.
- According to the normal distribution, 50% of the data falls above the mean.
- Therefore, approximately 50% of the 1000 students will have heights above 5'10".
4) Below 6':
- This range corresponds to below the mean.
- According to the normal distribution, 50% of the data falls below the mean.
- Therefore, approximately 50% of the 1000 students will have heights below 6'.
5) Above 5'8":
- This range corresponds to within 1 SD from the mean, but above 5'8".
- According to the normal distribution, 34% of the data falls within 1 SD from the mean in the positive direction.
- Therefore, approximately 34% of the 1000 students will have heights between 5'8" and 5'10", inclusive.
6) 5'8" - 6'4":
- This range corresponds to within 2 SDs from the mean, but between 5'8" and 6'4".
- According to the normal distribution, 81% of the data falls within 2 SDs from the mean.
- However, we need to subtract the probability of the range above 6'2" (which is 6%) from this percentage.
- Therefore, approximately 75% of the 1000 students will have heights between 5'8" and 6'4", inclusive.
Please note that these are approximate calculations based on the given normal distribution percentages and may vary slightly.
To find the approximate number of students in each height range, we will use the properties of the normal distribution and the given percentages for each standard deviation.
1) The range of 5'8" to 6' corresponds to a height range of 5'8" to 6'. This range falls within 1 standard deviation from the mean. According to the given hint, the area under the normal curve within 1 standard deviation is approximately 68%. Thus, approximately 68% of the students' heights will fall within this range.
2) The range of 5'6" to 6'2" corresponds to a height range of 5'6" to 6'2". This range falls within 2 standard deviations from the mean. According to the given hint, the area under the normal curve within 2 standard deviations is approximately 94%. Thus, approximately 94% of the students' heights will fall within this range.
3) The range above 5'10" corresponds to heights greater than 5'10". This range falls within 0 standard deviations or above from the mean. According to the given hint, the area under the normal curve within 0 standard deviations or above is approximately 50%. Thus, approximately 50% of the students' heights will fall within this range.
4) The range below 6' corresponds to heights less than 6'. This range falls within 1 standard deviation or below from the mean. According to the given hint, the area under the normal curve within 1 standard deviation or below is approximately 34%. Thus, approximately 34% of the students' heights will fall within this range.
5) The range above 5'8" corresponds to heights greater than 5'8". This range falls within 1 standard deviation or above from the mean. According to the given hint, the area under the normal curve within 1 standard deviation or above is approximately 84% (50% on one side plus half of the 68% on the other side). Thus, approximately 84% of the students' heights will fall within this range.
6) The range of 5'8" to 6'4" corresponds to a height range of 5'8" to 6'4". This range falls within 2 standard deviations or above from the mean. According to the given hint, the area under the normal curve within 2 standard deviations or above is approximately 98% (50% on one side plus 68% on the other side). Thus, approximately 98% of the students' heights will fall within this range.
By using the percentages provided and the properties of the normal distribution, we have estimated the approximate number of students in each height range.