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 52 young adults with pierced tongues. The researchers believed that it was reasonable to regard this sample as a random sample from the population of all young adults with pierced tongues. The researchers found receding gums, which can lead to tooth loss, in 18 of the study participants. a. Suppose you are interested in learning about the value of \(p,\) the proportion of all young adults with pierced tongues who have receding gums. This proportion can be estimated using the sample proportion, \(\hat{p} .\) What is the value of \(\hat{p}\) for this sample? b. Based on what you know about the sampling distribution of \(\hat{p}\), is it reasonable to think that this estimate is within 0.05 of the actual value of the population proportion? Explain why or why not. (Hint: See Example 8.4\()\)

Short answer

The sample proportion \(\hat{p} = 0.34615.\) With a calculated margin of error of 0.13637 (at the 95% confidence level), it can't be said with certainty that this estimate is within 0.05 of the actual value of the population proportion.

Step-by-step solution

Compute the Sample Proportion

The sample proportion, denoted as \(\hat{p}\), is calculated by dividing the number of successful outcomes (receding gums) by the total number of observations. For this sample, there were 18 cases of receding gums out of 52 participants. Therefore, \(\hat{p} = \frac{18}{52} = 0.34615 \) rounded to five decimal places.

Understanding Sampling Distribution of \(\hat{p}\)

The sampling distribution of \(\hat{p}\) can be approximated by a normal distribution if the sample size is large enough (usually n ≥ 30) and the population is much larger than the sample. Given these, it can be assumed that the sampling distribution of \(\hat{p}\) for this problem is approximately normally distributed.

Margin of Error

Usually a margin of error is paired with a confidence level, but this problem neglected to mention one. However, assuming a 95% confidence level, the margin of error (denoted as E) is calculated using the formula : \(E = Z\sqrt{\frac{\hat{p} (1 - \hat{p})}{n}}\). Then, is checked whether the estimate could reasonably be within this margin of error of the actual value. Here, z-value for 95% confidence level is approximately 1.96. Thus, \(E = 1.96 \sqrt{\frac{0.34615(1 - 0.34615)}{52}} = 0.13637\), rounded to five decimal places.

Reasonability Check

The calculated margin of error is more than 0.05. This means the estimate could most likely be more than 0.05 away from the actual population proportion. Therefore, it is not reasonable to believe the estimate is within 0.05 of the actual value of the population proportion, based on this sample.

Key concepts

Sample Proportion

Understanding the sample proportion is crucial in statistics as it represents an estimate of the population proportion based on a sample. In the textbook example, researchers examined a group of young adults with pierced tongues and used this group to estimate the proportion of all young adults with pierced tongues who have receding gums.

To calculate the sample proportion \( \hat{p} \), which is the fraction of the sample with the trait of interest, we take the number of 'successes' (participants with receding gums in this instance) and divide it by the total number of observations in the sample. The calculation in our example yields \( \hat{p} = \frac{18}{52} = 0.34615 \), indicating that roughly 34.615% of the sample participants exhibited receding gums.

It is important to comprehend that the sample proportion is an estimate of the true population proportion and its accuracy can be influenced by the size of the sample and how representative that sample is of the population.

Population Proportion

The population proportion refers to the true proportion of individuals in an entire population who have a particular characteristic. In our discussion of tongue piercings and tooth loss, the population proportion (denoted as \( p \)) is the actual percentage of all young adults with pierced tongues who have receding gums.

Since it's often not feasible to study an entire population, we rely on samples to estimate this proportion. Researchers use the sample proportion obtained from a study to make inferences about the population proportion. The accuracy of our estimation depends not only on the size of the sample but also on the randomness of the sample selection. If the sample is not representative, the estimate of the population proportion might be biased.

Margin of Error

The margin of error quantifies the extent to which we expect the sample proportion to vary from the true population proportion. It provides bounds within which we can be confident that the population proportion lies, based on our sample data.

The calculation for the margin of error typically involves a confidence level, which is the probability that the margin will contain the true population proportion. Although the original problem does not specify a confidence level, using a common 95% confidence level, we employ a standard formula involving the z-score that corresponds to this confidence level and the sample statistics.

The margin of error formula is \( E = Z \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \) where \( Z \) is the z-value from standard normal distribution, \( \hat{p} \) is the sample proportion, and \( n \) is the sample size. For our example, we calculated the margin of error to be 0.13637, which means we expect the true population proportion to be within this range above or below our sample proportion, with 95% confidence. If this margin is larger than what we consider acceptable (for instance, 0.05 in this scenario), we conclude that our estimate might not be sufficiently precise.

Normal Distribution

The normal distribution plays a fundamental role in the field of statistics, especially when analyzing sampling distributions. It is a continuous probability distribution that is symmetrical around the mean, describing a common pattern in which values cluster around a central peak.

When dealing with sample proportions, if the sample size is sufficiently large and the assumptions of randomness and independence of observations are met, the sampling distribution of the sample proportion can be approximated by a normal distribution. This approximation allows us to utilize z-scores and calculate margins of error, as illustrated in the textbook exercise.

In our example, we assume that the sampling distribution of the sample proportion \( \hat{p} \) for receding gums among young adults with pierced tongues can be approximated by a normal distribution. This is justified because the sample size is greater than 30, suggesting the central limit theorem is applicable, and thereby allowing the use of normal distribution properties for inference.