Answer :
To tackle this problem, we need to calculate two key statistics: the mean (X) and the standard deviation (S) for the given body temperatures. We will also need to determine the number of students and calculate the proportion percentage of students falling within one, two, and three standard deviations of the mean. Let's proceed step-by-step:
Step 1: Calculate the Mean (X)
The mean is calculated by adding all the data points together and then dividing by the number of data points.
Given temperatures: [tex]95.5, 95.8, 96.0, 98.6, 97.9, 95.9, 97.3, 95.9, 96.8, 97.2, 97.6, 99.1, 96.0, 96.4, 97.2, 97.5, 98.7, 98.2, 94.9, 96.0, 96.9, 97.8, 97.5, 97.8, 96.2, 96.9, 98.7, 97.8, 96.9, 99.1, 98.7, 96.1, 97.2, 99.0, 98.2, 96.3, 96.2, 98.2, 97.4, 98.4, 98.6, 97.4, 98.6[/tex]
Sum all temperatures: [tex]\sum_{i=1}^{n} \text{Temperature}_i = 4120.1[/tex]
Divide by the number of temperatures (n = 43):
[tex]X = \frac{4120.1}{43} \approx 95.8.57[/tex]
Step 2: Calculate the Standard Deviation (S)
Standard deviation is a measure of the amount of variation or dispersion of a set of values. It is calculated using the formula:
[tex]S = \sqrt{\frac{\sum_{i=1}^{n} (X_i - X)^2}{n - 1}}[/tex]
Calculate each temperature's deviation from the mean, square it, and sum these values:
[tex]\sum_{i=1}^{n} (X_i - X)^2 \approx 45.519[/tex]
Divide by [tex]n - 1[/tex]:
[tex]\frac{45.519}{42} \approx 1.083[/tex]
Take the square root:
[tex]S \approx \sqrt{1.083} \approx 1.041[/tex]
Step 3: Determine the Number of Students
Simply count the number of data points. Here, there are 43 body temperatures recorded, so there are 43 students.
Step 4: Proportion of Students within [tex]X±S[/tex], [tex]X±2S[/tex], and [tex]X±3S[/tex]
Calculate the boundaries for each interval and determine the number of observations within those limits.
[tex]X±S[/tex]: Calculate the range [tex](X-S, X+S)[/tex]
- Lower bound: [tex]98.57 - 1.041 = 97.53[/tex]
- Upper bound: [tex]98.57 + 1.041 = 99.61[/tex]
Count the number of temperatures within this range.
[tex]X±2S[/tex]: Calculate the range [tex](X-2S, X+2S)[/tex]
- Lower bound: [tex]98.57 - 2(1.041) = 96.49[/tex]
- Upper bound: [tex]98.57 + 2(1.041) = 100.65[/tex]
Count the number of temperatures within this range.
[tex]X±3S[/tex]: Calculate the range [tex](X-3S, X+3S)[/tex]
- Lower bound: [tex]98.57 - 3(1.041) = 95.44[/tex]
- Upper bound: [tex]98.57 + 3(1.041) = 101.69[/tex]
Count the number of temperatures within this range.
Finally, calculate the proportion by dividing the number of students within each interval by the total number of students (43) and multiply by 100 to get the percentage.