Question

Two machines P and Q are used to produce bags of cement of masses in kilogrammes shown in the table.

Machine P

50

51

48

50

51

52

54

51

51

Machine Q

54

49

56

47

50

51

52

53

Test if there is a difference between the two machines.

Answer 1

$H_0:\mu _1=\mu _2\\H_1:\mu _1\ne \mu _2\\\\n_1=9\\n_2=8\\\bar{x}_1=\frac{\sum{x_i}}{n_1}=50.8889\\\bar{x}_2=\frac{\sum{y_i}}{n_2}=51.5\\{s_1}^2=\frac{\sum{\left( x_i-\bar{x}_1 \right) ^2}}{n_1-1}=2.6111\\{s_2}^2=\frac{\sum{\left( y_i-\bar{x}_2 \right) ^2}}{n_2-1}=8.2857\\\nu =\frac{\left( \frac{{s_1}^2}{n_1}+\frac{{s_2}^2}{n_2} \right) ^2}{\frac{1}{n_1-1}\left( \frac{{s_1}^2}{n_1} \right) ^2+\frac{1}{n_2-1}\left( \frac{{s_2}^2}{n_2} \right) ^2}=\frac{\left( \frac{2.6111}{9}+\frac{8.2857}{8} \right) ^2}{\frac{1}{8}\left( \frac{2.6111}{9} \right) ^2+\frac{1}{8}\left( \frac{8.2857}{8} \right) ^2}=12.1558\approx 12\\T=\frac{\bar{x}_1-\bar{x}_2}{\sqrt{\frac{{s_1}^2}{n_1}+\frac{{s_2}^2}{n_2}}}=\frac{50.8889-51.5}{\sqrt{\frac{2.6111}{9}+\frac{8.2857}{8}}}=-0.530723\\P-value:\\P\left( \left| T \right|>0.5307 \right) =2F_{t,12}\left( -0.5307 \right) =2\cdot 0.3027=0.6054$