Setting Up a Test for a Mean

A fill label gives a claim to test

• A fictional random sample of 16 containers has mean fill 496 mL and sample standard deviation 8 mL. The label claims a population mean of 500 mL.
• The question is whether the population mean is below 500 mL. The difference in sample means alone does not settle that question.

Write hypotheses about the population

• Set $H_0:\mu=500$ and the preplanned lower-sided alternative $H_a:\mu<500$. The symbol $\mu$ names the population mean.
• Do not put the observed sample mean into a null hypothesis about the population. Do not change the direction because the sample happens to fall below the label.

Check when this reference model is useful

• Check random sampling and independence, including the 10% condition when sampling without replacement. With 16 observations, inspect skew and outliers carefully.
• The t model is exact for independent normal observations and can be approximate in suitable other cases. Passing an arithmetic check does not establish the sampling assumptions.

Standardize using sample variability

• Use $t=(\bar x-\mu_0)/(s/\sqrt n)$. Here $SE=8/\sqrt{16}=2$ mL.
• The observed statistic is $t=(496-500)/2=-2$. The t reference distribution has $df=16-1=15$.

Select the tail that answers the question

• The lower-sided alternative uses the area below $t=-2$ in the t distribution with 15 degrees of freedom. It is approximately 0.0320.
• The standard normal left-tail area at -2 is about 0.0228, a different value. Substituting a normal plot for this t reference changes the displayed evidence.

Report evidence rather than certainty

• At a prechosen significance level of 0.05, the approximate p-value 0.0320 leads to rejection of the 500 mL null claim in favour of a lower population mean.
• State the population, design, statistic, degrees of freedom, p-value and decision. The conclusion is not that every container is underfilled or that the true mean is certainly exactly 496 mL.

Write from a test statistic of negative 5

• The null mean is 500 grams.
• The sample mean is 490, the sample standard deviation is 8, and n is 16. The alternative is less than 500. The lesson statistic negative 2 is another sample.