# Investigate the significance level $α = 0.01$.

If you throw a coin in a vending machine, the coin is being weighed by the machine to determine its value.
For statistical purposes, you decide to throw $$4$$ fifty-cent coins in this machine and let $$\overline{X}_4$$ be the estimator for the mean weight $$μ$$ of a fifty-cent coin.
Assume that a single measurement has a normal distribution with expectation $$μ$$ and variance $$σ^2 = 0.04$$.
You find out that $$\overline{x}_4 = 7.54$$.
Investigate by using a statistical test whether $$μ$$ equals $$7.5$$ at significance level $$α = 0.01$$.

Solve:
I assume $$H_0:\mu=7.5$$ and $$H_1:\mu\neq7.5$$ So it is a two-tailed problem and I have an $$\alpha=0.005$$ on the right and the same on the left. So from the table the critical values for these $$\alpha$$ are 2.57 and -2.57. So I can compute $$Z=\frac{7.48-7.5}{\frac{\sqrt{0.004}}{\sqrt{4}}}=-0.63$$ that is inside the region where we don't reject $$H_0$$ so we don't reject it.

You have written the population variance as $$\sigma^2 = 0.004,$$ not $$0.04.$$ Also, in the problem you state that $$\bar X = 7.54,$$ but in your computation you have $$7.48.$$ Otherwise, you seem to be on the right track.

After fixing your typos, you should have $$Z = 0.4.$$ Then at level $$\alpha = 0.01 = 1\%,$$ you would reject if $$|Z| > 2.576,$$ so you cannot reject.

The P-value is the area under the standard normal curve outside the interval $$(-.4,.4),$$ which is $$0.6892 > 0.01,$$ and so (again) no rejection.

My computations from R statistical software are shown below:

(7.54-7.50)/ sqrt(.04/4)
[1] 0.4
qnorm(.995)
[1] 2.575829
2*pnorm(-.4)
[1] 0.6891565


Also, Minitab statistical software gives the following relevant output:

One-Sample Z

Test of μ = 7.5 vs ≠ 7.5
The assumed standard deviation = 0.2

N   Mean  SE Mean      99% CI         Z      P
4  7.540    0.100  (7.282, 7.798)  0.40  0.689


Because your hypothetical mean $$\mu = 7.5$$ is included in the 99% confidence interval, you would not reject at the 1% level.

• Ohh yes, I see my errors, thank you, so I'll reject the null hypothesis when $Z>0.4$ and when $Z<-0.4$, because both are critical values, correct? And the p-value should be $P(Z>0.4)+P(Z<-0.4)$? – Mark Jacon Jan 27 at 11:38
• Both comments correct. Also reject at 1% if 99% CI does not include hypothetical mean. 99% CI when $\sigma$ is known is of form $\bar X \pm 2.576\sigma/\sqrt{n}.$ Try it and compare with CI in Minitab output. – BruceET Jan 27 at 19:29