Question

Topical Painkiller Ointment The use of topical painkiller ointment or gel rather than pills for pain relief was approved just within the last few years in the US for prescription use only. \({ }^{13}\) Insurance records show that the average copayment for a month's supply of topical painkiller ointment for regular users is \$30. A sample of \(n=75\) regular users found a sample mean copayment of \(\$ 27.90\). (a) Identify each of 30 and 27.90 as a parameter or a statistic and give the appropriate notation for each. (b) If we take 1000 samples of size \(n=75\) from the population of all copayments for a month's supply of topical painkiller ointment for regular users and plot the sample means on a dotplot, describe the shape you would expect to see in the plot and where it would be centered. (c) How many dots will be on the dotplot you described in part (b)? What will each dot represent?

Short answer

(a) The average co-payment of 30 is a parameter denoted by \(\mu\) and the sample mean co-payment of 27.90 is a statistic denoted by \(\overline{X}\).\n (b) The plot would be approximately normally distributed and centered around the population mean, \(\mu\)=30. \n(c) There will be 1000 dots on the dotplot, each representing the average co-payment of a sample of 75 users.

Step-by-step solution

Identification

Firstly, let's identify the parameters and statistics given in the problem. The value 30 is the average co-payment of the population, making it a parameter. We will denote this parameter by \(\mu\). The value 27.90 is the average co-payment from a sample of 75 users, so it's a statistic. We will denote this statistic by \(\overline{X}\).

Expected Shape and Centre of Dotplot

Next, if we draw 1000 samples each of size 75 and plot the sample means on a dot plot, according to the Central Limit Theorem (given a large enough sample size), we can expect the shape of the plot to come out as approximately normal. As for where it's centered, the dotplot of sample means would be centered around the population mean, which is 30 in this case.

Dot Interpretation

In terms of plot interpretation, as we have made 1000 samples, each giving us one average co-payment, there will be 1000 dots on the plot. Each dot represents the average co-payment of a sample of 75 users.

Key concepts

Central Limit Theorem

The Central Limit Theorem (CLT) is a fundamental principle in statistics that provides a bridge between the world of data and probability. It states that, regardless of the population's distribution, the distribution of sample means will tend to be normal (or bell-shaped) as the sample size becomes large. This is incredibly useful because it allows statisticians to make inferences about population parameters using the normal distribution, which is well understood and for which tools and formulas are readily available.

The Central Limit Theorem has important implications for statistical practice. For instance, even if the original population of data is skewed, the distribution of the sample means will become more symmetric and more like a normal distribution as the sample size increases. It's critical to note, however, that the CLT applies when the sample size is 'large enough'. In practice, a sample size of 30 or more is often considered sufficient for the CLT to hold, although this can vary based on the population distribution's characteristics.

In the context of the exercise involving the cost of topical painkiller ointment, when we take many samples and calculate their means, the distribution of these sample means is predicted by the CLT to be approximately normal. This is why, if plotting the means of 1000 samples, the shape of the dotplot is expected to be bell-shaped and centered around the population mean.

Parameter vs. Statistic

Understanding the difference between a parameter and a statistic is crucial in statistics education. A parameter is a characteristic or measure that describes an aspect of an entire population. In contrast, a statistic describes a characteristic of a sample, a subset of the population. These concepts are foundational because they distinguish between universal truths (parameters) and observed truths (statistics), which are subject to sampling variability.

Parameters are fixed values but are usually unknown and often difficult to compute in reality due to the sheer size or accessibility of a population. Therefore, we use samples to estimate them. On the other hand, a statistic, since it is based on a sample, can be calculated directly but can vary from one sample to another. This variability is described by the sampling distribution of the statistic.

In our exercise example, the average co-payment of \(30 for the entire user population is a parameter (usually denoted by Greek letters, like \(\mu\)), while the average co-payment of \)27.90, obtained from a sample of 75 users, is a statistic (denoted by Roman letters like \(\overline{X}\)). Identifying these correctly is key to proper interpretation and application of statistical methods.

Sampling Distribution

The sampling distribution is a term that describes the distribution of a statistic, like the mean or proportion, that one would obtain from many samples of a specific size from a particular population. This distribution provides a picture of the variability of the statistic across different samples and is central to inferential statistics, which involves making conclusions about populations based on samples.

Each sample statistic serves as a single point in the sampling distribution, and the shape, center, and spread of this distribution provide important information about the statistic's reliability and precision as an estimator of the population parameter. For large sample sizes, as mentioned in the Central Limit Theorem, the sampling distribution of the sample mean will tend to be normally distributed regardless of the shape of the population distribution.

In the exercise example, for 1000 samples of size 75, the dotplot of the sample means would represent the sampling distribution of the mean co-payment. Each dot on this plot represents a sample mean, and the entire plot illustrates the variability of these means. If the sample size is large enough, which in this case it is, the dotplot would show a normal distribution centered around the population mean of $30.