Problem 49 - An Introduction to Mathematical Statistics and Its Applications

Question

Flonase is a nasal spray for diminishing nasal allergic symptoms. In clinical trials for side effects, 782 sufferers from allergic rhinitis were given a daily dose of \(200 \mathrm{mcg}\) of Flonase. Of this group, 126 reported headaches. A group of 758 subjects were given a placebo, and 111 of them reported headaches. Find a \(95 \%\) confidence interval for the difference in proportion of headaches for the two groups. Does the confidence interval suggest a statistically significant difference in the frequency of headaches for Flonase users?

Step-by-Step Solution

Answer

95% confidence interval for the difference in proportion of headaches is (0.01468, 0.01468), suggesting a statistically significant difference in the frequency of headaches for Flonase users.

Step 1: Calculate Proportions

First, we calculate the proportion of headaches in both groups. For the Flonase users, that is \(p1 = 126/782 = 0.16112\). For the placebo users, it is \(p2 = 111/758 = 0.14644\).

Step 2: Compute Standard Error

Next, we need to compute the standard error of the difference of proportions. The formula for this depends on the two proportions, \(p1\) and \(p2\), and the sizes of the two groups, \(n1 = 782\) and \(n2 = 758\). The formula is \(\sqrt{{{p1*(1-p1)}}/{n1} + {{p2*(1-p2)}}/{n2}}\), which gives us a standard error of \(0.01785\).

Step 3: Compute Confidence Interval

Now we can compute the 95% confidence interval for the difference in proportions. The formula is \((p1 - p2) \pm z*SE\), where \(z\) is the z-value for the desired confidence level (in this case, 1.96 for a 95% confidence level), and \(SE\) is the standard error. Using this formula we find that the 95% confidence interval = \((0.01468, 0.01468)\).

Step 4: Interpret Results

The final step is to interpret our results. Because the 95% confidence interval does not contain 0, this suggests a statistically significant difference in the frequency of headaches for the two groups.

Key Concepts

Statistical Significance

Statistical significance is a term used to determine if the difference observed between two or more groups in an experiment is due to something other than random chance. In other words, it helps us understand if the observed effects are real or if they could just be a result of sampling variability.

When conducting hypothesis testing in statistics, we use a significance level (commonly denoted as \(\alpha\)) to decide whether to reject the null hypothesis. The null hypothesis usually posits that there is no effect or no difference, while the alternative hypothesis suggests that there is an effect or a difference. A significance level of 0.05, for example, means that there is a 5% risk of concluding that a difference exists when there is none.

In our exercise about Flonase and the placebo group, we are looking for evidence that there is a true difference in the proportion of headaches reported. If our confidence interval does not include the value of zero, we can say that there is statistical significance. This is because a confidence interval that doesn't contain zero suggests that the observed difference is not likely to be zero and hence not due to random chance.

Proportion Difference

The difference in proportion is an important concept in statistics, especially when comparing two groups. It's quite simply the subtraction of one group's proportion from another's. In our example with Flonase and the placebo group, we first found the proportion of people reporting headaches after using Flonase and those using a placebo.

To calculate each group's proportion, we divide the number of subjects reporting headaches by the total number of subjects in that group. The difference in these proportions (\(p1 - p2\)) is what we are interested in. When we have the proportions from each group, we can subtract one from the other to get the difference in proportion. If this difference is considerably far from zero, it may suggest a meaningful difference in effect between the two groups.

Knowing the difference in proportions alone isn't enough, though. We must also understand how certain we are of this difference, which is where the confidence interval comes into play. The confidence interval gives us a range that likely contains the true difference in proportions if we were to take many samples.

Standard Error Calculation

Standard error is crucial in the calculation of confidence intervals as it gauges the accuracy of the sample mean compared to the population mean. Put simply, it estimates the variability that occurs by chance because we're using a sample of the population rather than the entire population.

In the case of difference in proportions, the standard error helps us understand how much the estimate of the difference between groups might vary from one sample to another. The formula incorporates the proportions of both groups and their respective sample sizes. As demonstrated in our Flonase exercise, calculating the standard error involves a formula \( SE = \sqrt{\frac{p1(1-p1)}{n1} + \frac{p2(1-p2)}{n2}} \), where \(p1\) and \(p2\) are the observed proportions and \(n1\) and \(n2\) are the sample sizes.

The 'standard error of the difference in proportions' provides us with a value that, when multiplied by the z-value for our confidence level, gives us the margin of error. This margin, added to and subtracted from our proportion difference, gives us the confidence interval which informs us about the precision of our estimate of the difference in proportions.