computing p-value with small n

Question

As part of the quality-control program for a catalyst manufacturing line, the raw materials (alumina and a binder) are tested for purity. The process requires that the purity of the alumina be greater than 85%. A random sample from a recent shipment of alumina yielded the following results (in %): $$ 93.2, 87.0, 92.1, 90.1, 87.3, 93.6$$ Using that information, I get that $$n = 6, \bar X =.91, S = 0.03$$ A) State the appropriate null and alternative hypotheses. I have that $$H_0:\mu_0<.85\space H_1:\mu_0\le.85$$ B)Compute the p-value. $$test\space statistic\space t=\frac{\sqrt{n}(\bar X - \mu_0)}{\sigma}=\frac{\sqrt{6}(.91-.85)}{.03} =4.9$$ Since n is sufficiently small, we calculate the t-distribution Assuming a 5% significance level, and with 5 degrees of freedom $t_{.025,5} =2.571$ C)Should the shipment be accepted? Explain. The shipment should not be accepted because 2.571< 4.9 which disproves the null hypothesis. We therefore accept the alternative hypothesis

Answer 1

$a)$: The null and alternative hypothesis should be:

$H_0: \mu = 0.85, H_1: \mu > 0.85$

$b)$: It is not necessary to get an exact $P$-value, you can estimate it with $df = n-1 = 6-1 = 5, \alpha = 0.05$, and the test is a one-tail test ( right tail ), then $P$-value $< 0.005 < 0.05 = \alpha $ because $4.9 > 4.032$ which corresponds to the $\alpha = 0.005$ using Triola textbook. Thus we reject $H_0$