Question

The positive effect of water fluoridation on dental health is well documented. One study that validates this is described in the article "Impact of Water Fluoridation on Children's Dental Health: A Controlled Study of Two Pennsylvania Communities" (American Statistical Association Proceedings of the Social Statistics Section \([1981]:\) 262-265). Two communities were compared. One had adopted fluoridation in 1966, whereas the other had no such program. Of 143 randomly selected children from the town without fluoridated water, 106 had decayed teeth, and 67 of 119 randomly selected children from the town with fluoridated water had decayed teeth. Let \(\pi_{1}\) denote the true proportion of children drinking fluoridated water who have decayed teeth, and let \(\pi_{2}\) denote the analogous proportion for children drinking unfluoridated water. Estimate \(\pi_{1}-\pi_{2}\) using a \(90 \%\) confidence interval. Does the interval contain \(0 ?\) Interpret the interval.

Short answer

The solution will be the resulting confidence interval which will be found after these calculations. If the confidence interval contains zero, that suggests that there may be no significant difference in decay rates between the two communities. The specific confidence interval will give more details about the significance and range of this difference.

Step-by-step solution

Calculate proportions

Firstly, the proportion of children with decayed teeth who drink fluoridated and unfluoridated water should be calculated. The proportion \(p_{1}\) of children drinking fluoridated water who have decayed teeth is \(67/119\). The corresponding proportion \(p_{2}\) for children drinking unfluoridated water is \(106/143\).

Calculate the difference in proportions

Next, find the difference in proportions, \(p_{1} - p_{2}\). This gives the estimated difference in the proportions of children with decayed teeth between those who drink fluoridated and unfluoridated water.

Compute Standard Error

The Standard Error (SE) for the difference in proportions estimate can be obtained by the formula: \(SE=\sqrt{\frac{p_{1}(1-p_{1})}{n_{1}}+\frac{p_{2}(1-p_{2})}{n_{2}}}\), where \(n_{1}\) and \(n_{2}\) are the number of observations, in this case the number of children in each community.

Find z-value and compute Confidence Interval

For a 90% confidence interval, the z-value will be \(Z=1.645\). This can be found from the standard normal distribution table or a calculator. The formula for confidence interval (CI) is given by: \(CI = (p_{1} - p_{2}) \pm Z*SE\). Now, plug in values to find the confidence interval.

Check Zero in Interval and Interpret Result

Now, check the confidence interval for zero. If the confidence interval contains zero, it implies that there is no significant difference between the proportions. Otherwise, one could say there's a significant difference. An interpretation of the interval is then provided.

Key concepts

Understanding Proportion in Statistics

In statistics, the term 'proportion' refers to the fraction of the total that possesses a certain attribute or characteristic. It's a measure of how prevalent a certain feature is within a specific group or population. Proportions are widely used in studies to make statements about larger populations based on sample data.

In the exercise example, proportions are used to determine the percentage of children with decayed teeth in communities with and without fluoridated water. By calculating these proportions, we can compare the dental health outcomes of the two different environments. This kind of analysis is critical not only in health studies but in various fields where we compare the occurrences of certain characteristics among groups.

When interpreting proportions, it's crucial to understand that they represent a snapshot based on the sample data at hand. In the given study, the proportions of children with decayed teeth—calculated as the number of children with decayed teeth divided by the total number of children surveyed in each community—serve as estimates for what we might expect in the broader population of similar communities.

Standard Error: Estimating the Variability

Standard Error (SE) is a statistical measure that indicates the accuracy with which a sample represents a population. In simple terms, it's a gauge of how much the sample mean would vary if you drew different samples from the same population.

The standard error of the difference in proportions, as seen in our exercise, tells us how precise our estimate of that difference is. The formula \(SE=\sqrt{\frac{p_1(1-p_1)}{n_1}+\frac{p_2(1-p_2)}{n_2}}\) accounts for variability within each group of children—those drinking fluoridated and unfluoridated water. The smaller the standard error, the more reliable our estimated difference in proportion is.

Why is SE important? Because it plays a critical role in constructing confidence intervals and conducting hypothesis tests. Without a reliable SE, we can't trust the accuracy of these intervals or tests. Thus, when interpreting the standard error, comprehension that a smaller SE indicates a more precise estimate is fundamental.

Statistical Significance and Its Relevance

Statistical significance is a determination of whether an observed effect or difference is likely to be genuine or if it could have happened by random chance. This concept is central to hypothesis testing and can provide a strong foundation for making inferences in research.

In the context of our exercise, we're looking at whether the difference in tooth decay between children in communities with and without fluoridated water is statistically significant. After calculating the confidence interval, we check to see if zero is within this interval. If it is not, we infer that the difference in proportions is statistically significant—that is, it's unlikely to be due to just chance alone.

Interpreting the interval is crucial: A confidence interval that doesn't include zero suggests a meaningful difference that could warrant further investigation or action, whereas an interval containing zero might indicate that any observed differences could simply be the result of random variation. Understanding statistical significance allows students and researchers to make better judgments about their data and to draw more accurate conclusions from their studies.