Respuesta :
Answer:
The 99% confidence interval is (-20.774, 2.774)
Step-by-step explanation:
The 60 seconds and 120 seconds breaking strength measures are;
60 seconds: 43, 52, 52, 58, 49, 52, 41, 52, 56, 51
120 seconds: 59, 55, 59, 66, 62, 55, 57, 66, 66, 51
The Mean and standard deviation for the data are obtained as follows;
The mean strength for the 60 seconds treatment samples, μX = 50.6 N
The standard deviation for the 60 seconds treatment samples, sX = 5.211099 N
The number of threads in the sample, n₁ = 10
The mean strength for the 60 seconds treatment samples, μY = 59.6 N
The standard deviation for the 60 seconds treatment samples, sY = 5.295701 N
The number of threads in the sample, n₂ = 10
The 99% confidence interval for the difference in mean, is given as follows;
[tex]\left (\mu X- \mu Y \right )\pm t_{\left(\dfrac{\alpha}{2}, df \right) } \cdot \sqrt{\dfrac{sX^{2}}{n_{1}}+\dfrac{sY^{2}}{n_{2}}}[/tex]
The degrees of freedom, df = n₁ + n₂ - 2 = 20 - 2 = 18
(1 - α)·100 = 99%
α = 1 - 99/100 = 0.01
α/2 = 0.01/2 = 0.005
The critical-t at 0.005 significant level and df = 18 is 2.878
The confidence interval is therefore;
[tex]C.I. = \left (50.6- 59.6 \right )\pm 2.878 \right) } \times \sqrt{\dfrac{5.211099^{2}}{10}+\dfrac{5.295701^{2}}{10}}[/tex]
C.I. = -9 ± 11.774425
∴ By rounding to three decimal places, the 99% confidence interval is (-20.774, 2.774).
