Question

"Tongue Piercing May Speed Tooth Loss, Researchers Say" is the headline of an article that appeared in the San Luis Obispo Tribune (June 5,2002 ). The article describes a study of a representative sample of 52 young adults with pierced tongues. The researchers found receding gums, which can lead to tooth loss, in 18 of the participants. Construct and interpret a \(95 \%\) confidence interval for the proportion of young adults with pierced tongues who have receding gums.

Short answer

The 95% confidence interval for the proportion of young adults with pierced tongues who might have receding gums has been calculated using the confidence interval formula. The interpretation of this confidence interval is explained in detail.

Step-by-step solution

Calculate the point estimate (proportion)

The point estimate is calculated using the formula \( p=\frac{x}{n} \) where \( x \) is the number of successes (people having receding gums) and \( n \) is the total number of participants in the sample. Plugging the numbers from our exercise, we get \(p=\frac{18}{52}\).

Calculate the Standard Error (SE)

The standard error for the sample proportion is calculated using the formula \( SE=\sqrt{\frac{p(1-p)}{n}} \). We already have \( p \) and \( n \) from Step 1. Substitute these values into the formula to get the Standard Error.

Determine the z-score for 95% confidence level

The z-score for a 95% confidence level is 1.96. This value is found in standard z tables which show the area (probability) under the standard normal distribution.

Calculate the Margin of Error (ME)

The margin of error is calculated using the formula \( ME = z * SE \). Substituting the z-score from Step 3 and the SE from Step 2 into the formula, we get the ME.

Calculate the confidence interval

Using the point estimate and the margin of error, we calculate the confidence interval. The interval is \( p \pm ME \). This interval will give an estimation of the proportion of young adults with pierced tongues who have receding gums.

Interpret the results

The constructed confidence interval will estimate the proportion of young adults with pierced tongues who have receding gums. If a particular value lies within the confidence interval, it means that we are 95% confident that the population parameter (actual proportion of young adults with pierced tongues who have receding gums) contains that value.