Question

compute the mean, range, and standard deviation for the data items in each of the three samples. then describe one way in which are different
sample a: 27, 33, 39, 45, 51, 57, 63
sample b: 27, 30, 33, 45, 57, 60, 63
sample c: 27, 27, 27, 45, 63, 63, 63
in which of the following ways are the samples alike?
standard deviation
range
mean
mean and range
in which of the following ways are the samples different?
range and standard deviation
mean and standard deviation
standard deviation
mean and range

Explanation:

Step1: Calculate the range

The range is calculated as \(Range = Max - Min\).
For Sample A: \(Max = 63\), \(Min = 27\), so \(Range_A=63 - 27=36\)
For Sample B: \(Max = 63\), \(Min = 27\), so \(Range_B=63 - 27=36\)
For Sample C: \(Max = 63\), \(Min = 27\), so \(Range_C=63 - 27=36\)

Step2: Calculate the mean

The mean formula is \(\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}\)
For Sample A: \(n = 7\), \(\sum x_i=27 + 33+39 + 45+51+57+63=315\), \(\bar{x}_A=\frac{315}{7} = 45\)
For Sample B: \(n = 7\), \(\sum x_i=27+30 + 33+45+57+60+63=315\), \(\bar{x}_B=\frac{315}{7}=45\)
For Sample C: \(n = 7\), \(\sum x_i=27+27+27+45+63+63+63=315\), \(\bar{x}_C=\frac{315}{7}=45\)

Step3: Calculate the standard deviation

The formula for the sample standard deviation is \(s=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}}\)
For Sample A:
\((27 - 45)^2+(33 - 45)^2+(39 - 45)^2+(45 - 45)^2+(51 - 45)^2+(57 - 45)^2+(63 - 45)^2\)
\(=(- 18)^2+(-12)^2+(-6)^2+0^2+6^2+12^2+18^2\)
\(=324 + 144+36+0+36+144+324=1008\)
\(s_A=\sqrt{\frac{1008}{6}}=\sqrt{168}\approx12.96\)
For Sample B:
\((27 - 45)^2+(30 - 45)^2+(33 - 45)^2+(45 - 45)^2+(57 - 45)^2+(60 - 45)^2+(63 - 45)^2\)
\(=(-18)^2+(-15)^2+(-12)^2+0^2+12^2+15^2+18^2\)
\(=324+225 + 144+0+144+225+324=1386\)
\(s_B=\sqrt{\frac{1386}{6}}=\sqrt{231}\approx15.20\)
For Sample C:
\((27 - 45)^2+(27 - 45)^2+(27 - 45)^2+(45 - 45)^2+(63 - 45)^2+(63 - 45)^2+(63 - 45)^2\)
\(=(-18)^2+(-18)^2+(-18)^2+0^2+18^2+18^2+18^2\)
\(=324+324+324+0+324+324+324 = 1944\)
\(s_C=\sqrt{\frac{1944}{6}}=\sqrt{324}=18\)

Answer:

The samples are alike in terms of Mean and Range. The samples are different in terms of Standard Deviation. So for the first question (In which of the following ways are the samples alike?), the answer is Mean and Range. For the second question (In which of the following ways are the samples different?), the answer is Standard Deviation.