Question

find the range and the standard deviation for the three samples below. sample a: 41, 43, 45, 47, 49, 51, 53 sample b: 41, 42, 43, 47, 51, 52, 53 sample c: 41, 41, 41, 47, 53, 53, 53

Explanation:

Step1: Calculate the range

The range is calculated as \(Range = Max - Min\).
For Sample A: \(Max = 53\), \(Min = 41\), so \(Range=53 - 41=12\).
For Sample B: \(Max = 53\), \(Min = 41\), so \(Range = 53-41 = 12\).
For Sample C: \(Max = 53\), \(Min = 41\), so \(Range=53 - 41=12\).

Step2: Calculate the mean \(\bar{x}\)

The formula for the mean of a sample \(x_1,x_2,\cdots,x_n\) is \(\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}\).
For Sample A: \(n = 7\), \(\sum_{i=1}^{7}x_i=41 + 43+45 + 47+49+51+53=329\), \(\bar{x}_A=\frac{329}{7}=47\).
For Sample B: \(n = 7\), \(\sum_{i = 1}^{7}x_i=41+42 + 43+47+51+52+53=309\), \(\bar{x}_B=\frac{309}{7}\approx44.14\).
For Sample C: \(n = 7\), \(\sum_{i=1}^{7}x_i=41+41+41+47+53+53+53=329\), \(\bar{x}_C=\frac{329}{7}=47\).

Step3: Calculate the variance \(s^{2}\)

The formula for the sample variance is \(s^{2}=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^{2}}{n - 1}\).
For Sample A:
\(\sum_{i=1}^{7}(x_i - 47)^{2}=(41 - 47)^{2}+(43 - 47)^{2}+(45 - 47)^{2}+(47 - 47)^{2}+(49 - 47)^{2}+(51 - 47)^{2}+(53 - 47)^{2}\)
\(=(-6)^{2}+(-4)^{2}+(-2)^{2}+0^{2}+2^{2}+4^{2}+6^{2}=36 + 16+4+0 + 4+16+36=112\)
\(s_{A}^{2}=\frac{112}{6}\approx18.67\)
For Sample B:
\(\sum_{i = 1}^{7}(x_i - 44.14)^{2}=(41-44.14)^{2}+(42 - 44.14)^{2}+(43 - 44.14)^{2}+(47 - 44.14)^{2}+(51 - 44.14)^{2}+(52 - 44.14)^{2}+(53 - 44.14)^{2}\)
\(\approx(-3.14)^{2}+(-2.14)^{2}+(-1.14)^{2}+2.86^{2}+6.86^{2}+7.86^{2}+8.86^{2}\)
\(\approx9.86+4.58+1.30+8.18+47.06+61.78+78.49=211.23\)
\(s_{B}^{2}=\frac{211.23}{6}\approx35.21\)
For Sample C:
\(\sum_{i=1}^{7}(x_i - 47)^{2}=3\times(41 - 47)^{2}+(47 - 47)^{2}+3\times(53 - 47)^{2}\)
\(=3\times(-6)^{2}+0^{2}+3\times6^{2}=3\times36+0 + 3\times36=216\)
\(s_{C}^{2}=\frac{216}{6}=36\)

Step4: Calculate the standard deviation \(s\)

The formula for the sample standard deviation is \(s=\sqrt{s^{2}}\).
For Sample A: \(s_A=\sqrt{18.67}\approx4.32\).
For Sample B: \(s_B=\sqrt{35.21}\approx5.93\).
For Sample C: \(s_C=\sqrt{36}=6\).

Answer:

	Range	Standard Deviation
Sample B	\(12\)	\(\approx5.93\)
Sample C	\(12\)	\(6\)