Question

iqr = 4
(b) make a frequency table using four classes.
class 6 - 8 9 - 11 12 - 14 15 - 17
f
then estimate the mean and sample standard deviation using the frequency table. (round s to two decimal places.)
x =
s =
compute a 75% chebyshev interval centered about the mean. (enter your answer in the form: lower limit to upper limit. include the word \to.\ round your numerical values to two decimal places.)
(c) if you have a statistical calculator or computer, use it to find the actual sample mean and sample standard deviation. (round s to four decimal places )
x =
s =
(d) based on this data, would you be surprised to find a plot with soil water content of 3?
yes
no

Explanation:

Step1: Determine class - midpoints

For class 6 - 8, mid - point $x_1=\frac{6 + 8}{2}=7$; for 9 - 11, $x_2=\frac{9+11}{2}=10$; for 12 - 14, $x_3=\frac{12 + 14}{2}=13$; for 15 - 17, $x_4=\frac{15+17}{2}=16$. But we need the frequencies first. Since they are not given in the problem statement, assume frequencies $f_1,f_2,f_3,f_4$. Let the total number of data points be $n=f_1 + f_2+f_3+f_4$.

Step2: Calculate the mean $\bar{x}$

The formula for the mean from a frequency table is $\bar{x}=\frac{\sum_{i = 1}^{4}f_ix_i}{n}=\frac{f_1x_1 + f_2x_2+f_3x_3+f_4x_4}{f_1 + f_2+f_3+f_4}$.

Step3: Calculate the sample standard deviation $s$

The formula for the sample standard deviation from a frequency table is $s=\sqrt{\frac{\sum_{i = 1}^{4}f_i(x_i-\bar{x})^2}{n - 1}}$.

Step4: Chebyshev's inequality

For a 75% Chebyshev interval, $k = 2$ (since $1-\frac{1}{k^2}=0.75$). The interval is $\bar{x}-ks$ to $\bar{x}+ks$.

Since we don't have the frequencies, we can't give numerical answers for the mean, standard - deviation and Chebyshev interval.

Answer:

Since frequencies are not given, we cannot calculate the mean, sample standard deviation and Chebyshev interval numerically. For part (d), if 3 is far outside the range of values considered in the frequency - table classes (6 - 17) and based on the data characteristics (if we assume the data is concentrated around the classes given), we would answer Yes.