Question

A paint company produces glow in the dark paint with an advertised glow time of 15 min. A painter is interested in finding out if the product behaves worse than advertised. She sets up her hypothesis statements as H0 : µ ≤ 15 and Ha : µ > 15, then calculates a test statistic of z = −2.30. What would be the conclusions of her hypothesis test at significance levels of α = 0.05, α = 0.01, and α = 0.001?

Answer 1

Answer:

$p_v = P(z<-2.30) =0.0107$

Now we can decide based on the significance level $\alpha$. If $p_v <\alpha$ we reject the null hypothesis and in other case we FAIL to reject the null hypothesis.

$\alpha=0.05$ we see that $p_v< \alpha$ so then we have enough evidence to reject the null hypothesis and we can conclude that the true mean is significantly less than 15

$\alpha=0.01$ we see that $p_v> \alpha$ so then we have enough evidence to FAIL to reject the null hypothesis and we can conclude that the true mean is NOT significantly less than 15

$\alpha=0.001$ we see that $p_v> \alpha$ so then we have enough evidence to FAIL to reject the null hypothesis and we can conclude that the true mean is NOT significantly less than 15

Explanation:

For this case they conduct the following system of hypothesis for the ture mean of interest:

Null hypothesis: $\mu \leq 15$

Alternative hypothesis: $\mu >15$

The statistic for this hypothesis is:

$z = \frac{\bar X -\mu}{\frac{\sigma}{\sqrt{n}}}$

And on this case the value is given $z = -2.30$

For this case in order to take a decision based on the significance level we need to calculate the p value first.

Since we have a lower tailed test the p value would be:

$p_v = P(z<-2.30) =0.0107$

Now we can decide based on the significance level $\alpha$. If $p_v <\alpha$ we reject the null hypothesis and in other case we FAIL to reject the null hypothesis.

$\alpha=0.05$ we see that $p_v< \alpha$ so then we have enough evidence to reject the null hypothesis and we can conclude that the true mean is significantly less than 15

$\alpha=0.01$ we see that $p_v> \alpha$ so then we have enough evidence to FAIL to reject the null hypothesis and we can conclude that the true mean is NOT significantly less than 15

$\alpha=0.001$ we see that $p_v> \alpha$ so then we have enough evidence to FAIL to reject the null hypothesis and we can conclude that the true mean is NOT significantly less than 15