Question

Consider the test H0: μ <=100 vs H1: μ >100 Suppose that a sample of size 36 has a sample mean of 105. (i) Determine the p-value of this outcome if the population standard deviation is known to be 15. (ii) Based on the p-value obtained in (i) above for what values of α you would Reject H0 (a) 0.01, (b) 0.02 (c) 0.06. You are not expected to perform the test all over again for each value of α; instead conclude based on outcome of (i) above.

Answer 1

Answer on Question #46750 – Math – Statistics and Probability

Consider the test $H_0: \mu \leq 100$ vs $H_1: \mu > 100$. Suppose that a sample of size $n = 36$ has a sample mean of $\bar{x} = 105$.

(i) Determine the p-value of this outcome if the population standard deviation is known to be $\sigma = 15$.

(ii) Based on the p-value obtained in (i) above for what values of $\alpha$ you would reject $H_0$

(a) 0.01,

(b) 0.02

(c) 0.06.

You are not expected to perform the test all over again for each value of $\alpha$; instead conclude based on outcome of (i) above.

Solution

(i)

$$z = \frac {\bar {x} - \mu}{\frac {\sigma}{\sqrt {n}}} = \frac {105 - 100}{\frac {15}{\sqrt {36}}} = 2.$$ $$p - value = P(z > 2) = 0.0228.$$

(ii) (a) $\alpha = 0.01$. This p-value is bigger than $\alpha$, thus we don't reject $H_0$.

(b) $\alpha = 0.02$. This p-value is bigger than $\alpha$, thus we don't reject $H_0$.

(c) $\alpha = 0.06$. This p-value is smaller than $\alpha$, thus we reject $H_0$.

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