9.1 Quick revision

We strongly recommend trying these Quick revision questions before your tutorial.

Researchers studied the impact of a gluten-free diet (needed by coeliacs) on dental cavities (Khalaf et al. 2020). Some of the summary information regarding the number of decayed, missing and filled teeth (DMFT) is shown below. The researchers wanted to compare the mean DMFT for coeliacs and non-coeliacs.

TABLE 9.1: Coeliacs and dental cavities.
	Sample size	Mean	Standard deviation	Standard error
Coeliac (C)	23	8.39	4.4	0.92
Non-coeliac (NC)	23	8.17	4.1	0.86
Difference		0.22		1.3

An exact \(95\)% CI is given as for the difference is \(-2.32\) to \(2.76\).

Using the \(68\)--\(95\)--\(99.7\) rule gives a slightly different CI. Why?
The researchers made a calculation error. The researchers calculated an exact \(95\)% CI (not an approximate \(95\)% CI). The statistical validity conditions are not met.

True or false: The difference is computed as the number of DMFT for coeliacs minus non-coeliacs.
True or false: One of the values for the CI is a negative value, which must be an error, since a negative number of DMFT is impossible.
We are \(95\)% confident that the difference between the population means is:
Larger for coeliacs Smaller for coeliacs Between 2.32 higher for non-coeliacs to 2.76 higher for coeliacs. Between 2.76 higher for non-coeliacs to 2.32 higher for coeliacs.

True or false: The null hypothesis is \(H_0\): \(\mu_{C} - \mu_{NC} = 0\).
True or false: The alternative hypothesis is \(H_0\): \(\mu_{C} - \mu_{NC} > 0\).
True or false: Since the two sample means are different, we reject the null hypothesis.
Suppose the difference between the means was computed as \(\mu_C - \mu_{NC}\). What does this measure?
How much larger the mean number of cavities are for non-coeliacs How much larger the mean number of cavities are for coeliacs

The researchers used an exact \(95\)%CI, which gives very similar answers to using an approximate \(95\)% CI. (The \(68\)--\(95\)--\(99.7\) rule gives approximate multipliers only.)
TRUE. If we take the coeliac mean minus the non-coeliac mean, we would have \(8.39 - 8.17 = 0.22\), which is what is given in the table.
FALSE. The CI is for the difference in the DMFT. The value is not about the number of teeth; it is about the difference in the number of teeth between coeliacs and non-coeliacs. A negative value is fine if we understand what the CI is estimating.
The positive value of \(2.76\) means (as we saw in the first Quick Revision question) that coeliacs have a mean of \(2.76\) more DMFT.
So the negative value of \(-2.32\) means that non-coeliacs have a mean of \(2.32\) more DMFT.
TRUE. Remember that all hypotheses are about population parameters, and have an "equals to" in there somewhere (the "no difference, no change, no relationship" position).
FALSE. The hypothesis is about population parameters (\(\mu\)), which is good.
But the actual question was only about studying the difference, which suggests a two-tailed alternative hypothesis (not a one-tailed alternative hypothesis as given).
FALSE. The sample means could be different for one of two reasons: (a) the population means are actually the same, but the sample means are just different due to sampling variation; or (b) because the population means are different.
We don't know which one of these reasons is the reason here.
How much larger the mean number of cavities are for coeliacs.

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References

Khalaf ME, Akbar A, Alkhubaizi Q, Qudeimat M. Caries among adult patients with controlled celiac disease: A cross-sectional study. Special Care in Dentistry. Wiley Online Library; 2020;40(5):457–63.