Question

In testing the strength of chains under heat treatments A and B, the following two sets of strengths were found.

A : 3610 , 3608 , 3672

B : 3416 , 3460 , 3390

a) Find a 90 % confidence interval for $\sigma_a ^2 /\sigma_b ^2$ . State your assumption(s) for your answers to be valid.

b) On the basis of the confidence interval in part (a), do you think it is likely that

$\sigma_a ^2 =\sigma_b ^2$ ?Why?

Answer 1

For A, s_a² = 1324

For B, s_b² = 1252

a) At 90% confidence intrval the critical values are F( $\small \alpha/2, n_{a} - 1, n_{b} - 1$ ) = 19

                                                                                 F( $\small 1 - \alpha/2, n_{a} - 1, n_{b} - 1$ ) = 0.0526

The 90% confidence interval for $\small \sigma_{a}^2/{\sigma_{b}^2}$ is

(s_a²/s_b²) * (1/F( $\small \alpha/2, n_{a} - 1, n_{b} - 1$ )) < $\small \sigma_{a}^2/{\sigma_{b}^2}$ < (s_a²/s_b²) * (1/F( $\small 1 - \alpha/2, n_{a} - 1, n_{b} - 1$ )

= (1324/1252) * (1/19) < $\small \sigma_{a}^2/{\sigma_{b}^2}$ < (1324/1252) * (1/0.0526 )

=   0.0557 < $\small \sigma_{a}^2/{\sigma_{b}^2}$ < 20.1047

b) Since the confidence interval cointans 1, so we can be conclude that $\small \sigma_{a}^2 = {\sigma_{b}^2}$