the heights of 1000 students in a college are normally distributed with a mean 5’10" and SD 2". Use 68% for the region from the mean to 1-SD on either side; 94% for the region from the mean to 2-SD on either side and 98% for the region from the mean to 3-SD on either side. Find the approximate number of students in each range of the heights:
28) 5’8"–6’
29) 5’6"–6’2"
30) Above 5’10"
31) Below 6’
32) Above 5’8"
33) 5’8"–6’4"
Z = # of SD away from the mean = (x - mean)/SD
Calculate the Z score values for each height. Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions desired. Multiply that proportion by 1000.
To find the approximate number of students in each range of heights, we can use the properties of the normal distribution and the given percentage values for different standard deviations.
First, let's convert the heights to a uniform unit, such as inches:
5'10" = (5 * 12) + 10 = 70 inches
2" standard deviation = 2 inches
Now, let's calculate the heights for the given ranges:
28) 5'8"–6':
To find the number of students within 1 standard deviation from the mean, we can use the 68% rule. This means that 68% of the students' heights will fall within the range of 1 standard deviation from the mean.
To calculate the lower range:
5'8" = (5 * 12) + 8 = 68 inches
To calculate the upper range:
6' = (6 * 12) = 72 inches
So, the range 5'8"-6' corresponds to the range 68-72 inches. Approximately 68% of the students fall into this range.
Hence, the approximate number of students in this range is 68% of 1000 = 0.68 * 1000 = 680 students.
29) 5'6"–6'2":
To find the number of students within 2 standard deviations from the mean, we can use the 94% rule. This means that 94% of the students' heights will fall within the range of 2 standard deviations from the mean.
To calculate the lower range:
5'6" = (5 * 12) + 6 = 66 inches
To calculate the upper range:
6'2" = (6 * 12) + 2 = 74 inches
So, the range 5'6"-6'2" corresponds to the range 66-74 inches. Approximately 94% of the students fall into this range.
Hence, the approximate number of students in this range is 94% of 1000 = 0.94 * 1000 = 940 students.
30) Above 5'10":
To find the number of students above the mean height (5'10"), we need to calculate the area to the right of the mean under the normal distribution curve. This corresponds to students whose heights are greater than 70 inches.
Since 68% falls within 1 standard deviation from the mean, the area to the right of the mean will be (100% - 68%) / 2 = 16% on each side.
Hence, the approximate number of students above 5'10" is 16% of 1000 = 0.16 * 1000 = 160 students.
31) Below 6':
To find the number of students below 6' (72 inches), we need to calculate the area to the left of the upper range under the normal distribution curve.
Since we already know that 68% falls within 1 standard deviation from the mean, the area to the left of the mean will be (100% - 68%) / 2 = 16% on each side.
Hence, the approximate number of students below 6' is 16% of 1000 = 0.16 * 1000 = 160 students.
32) Above 5'8":
To find the number of students above 5'8" (68 inches), we need to calculate the area to the right of this height under the normal distribution curve.
Since 68% falls within 1 standard deviation from the mean, the area to the right of 5'8" will be (100% - 68%) / 2 = 16% on each side.
Hence, the approximate number of students above 5'8" is 16% of 1000 = 0.16 * 1000 = 160 students.
33) 5'8"–6'4":
To find the number of students within 3 standard deviations from the mean, we can use the 98% rule. This means that 98% of the students' heights will fall within the range of 3 standard deviations from the mean.
To calculate the lower range:
5'8" = (5 * 12) + 8 = 68 inches
To calculate the upper range:
6'4" = (6 * 12) + 4 = 76 inches
So, the range 5'8"-6'4" corresponds to the range 68-76 inches. Approximately 98% of the students fall into this range.
Hence, the approximate number of students in this range is 98% of 1000 = 0.98 * 1000 = 980 students.
To summarize:
28) 5'8"–6': Approximately 680 students
29) 5'6"–6'2": Approximately 940 students
30) Above 5'10": Approximately 160 students
31) Below 6': Approximately 160 students
32) Above 5'8": Approximately 160 students
33) 5'8"–6'4": Approximately 980 students