To find the standard error of the sample mean using the given weights of the students, we need to follow these steps:
1. Calculate the Sample Mean ([tex]\( \bar{x} \)[/tex]):
To find the sample mean, sum up all the weights and divide by the number of students.
[tex]\[
\bar{x} = \frac{128 + 193 + 166 + 147 + 202 + 183 + 181 + 158}{8}
\][/tex]
Simplifying this calculation, we get:
[tex]\[
\bar{x} = \frac{1358}{8} = 169.75
\][/tex]
2. Calculate the Sample Standard Deviation (s):
The sample standard deviation measures the amount of variation or dispersion of a set of values. The formula for sample standard deviation is:
[tex]\[
s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}
\][/tex]
where [tex]\( x_i \)[/tex] represents each individual weight, [tex]\( \bar{x} \)[/tex] is the sample mean, and [tex]\( n \)[/tex] is the sample size.
Substituting the values, we get:
[tex]\[
s \approx 24.77
\][/tex]
3. Calculate the Sample Size (n):
In this case, the sample size [tex]\( n \)[/tex] is the number of students. There are 8 students.
[tex]\[
n = 8
\][/tex]
4. Calculate the Standard Error of the Sample Mean (SE):
The standard error of the sample mean is given by the formula:
[tex]\[
SE = \frac{s}{\sqrt{n}}
\][/tex]
Substituting [tex]\( s = 24.77 \)[/tex] and [tex]\( n = 8 \)[/tex], we get:
[tex]\[
SE = \frac{24.77}{\sqrt{8}}
\][/tex]
[tex]\[
SE \approx \frac{24.77}{2.83} \approx 8.76
\][/tex]
Therefore, the standard error of the sample mean, rounded to the hundredths place, is 8.76.
