The P-value for a two-sided test of the null hypothesis $H_0:\mu =10$is $0.06$

(a) Does the$95\%$confidence interval for$M$include $10$? Why or why not?

(b) Does the $90\%$confidence interval for$M$include $10$? Why or why not

Given in the question that, The P-value for a two-sided test of the null hypothesis $H_0:\mu =10$is $0.06$.

we need to find that Whether the$95\%$confidence interval for $\mu$include $10$.

$H_0:\mu =10$

$P-$value $=0.06$

Reject, $H_0$, if level of significance$\left(\alpha \right)<P-$value .

The null and alternative hypotheses could be written as:

$H_0:\mu =10$

$H_a:\mu \ne 10$

The level of significance is$0.05$

Here,

$p-$value $(0.06)>\alpha (0.05)$

Thus, the null hypothesis is not rejected. So, it could be concluded that $95\%$confidence interval for $\mu$will includes $10$

Given in the question that, The $P$-value for a two-sided test of the null hypothesis $H_0:\mu =10$is $0.06$.

We need to find that Whether the$90\%$confidence interval for $\mu$include$10$.

Here,

$H_0:\mu =10$

$P$-value$=0.06$

Reject, $H_0$, if level of significance $\left(\alpha \right)<P-$value

The null and alternative hypotheses could be written as:

$H_0:\mu =10$

$H_a:\mu \ne 10$

The level of significance is $0.10$.

Here, $p-$value $(0.06)<\alpha (0.10)$

Thus, the null hypothesis is rejected. So, it could not be concluded that $90\%$confidence interval for $\mu$ will include$10.$