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 int eh US for prescription use only. 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 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.
_ _ = 30 Parameter or statistic? ______________
_ _ = 27.90 Parameter or statistic? ______________
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
dot plot, which of the following best describes both the shape and where it would be
centered (choose one)?
(A) The distribution would be symmetric, bell-shaped, and centered at 27.90
(B) The distribution would be symmetric, bell-shaped, and centered at 30.
(C) The distribution would be skewed, unimodal, and centered at 27.90
(D) The distribution would be skewed, unimodal, and centered at 30.
c) How many dots will be on the dotplot described in (b)? _
d) What will each dot represent?
The correct answer and explanation is :
a) Identify each of 30 and 27.90 as a parameter or a statistic and give the appropriate notation for each.
- 30 = Parameter or Statistic?
- 30 = Parameter
- This is a population parameter, representing the average copayment for a month’s supply of the topical painkiller ointment for all users (across the population). Parameters describe a characteristic of the entire population.
- 27.90 = Parameter or Statistic?
- 27.90 = Statistic
- This is a sample statistic, representing the average copayment for a month’s supply of the ointment among the sample of 75 regular users. Statistics describe a characteristic of a sample, not the entire population.
b) Description of the Dot Plot and Centering
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 dot plot, the distribution of those sample means would best be described as:
- (B) The distribution would be symmetric, bell-shaped, and centered at 30.
Explanation:
- According to the Central Limit Theorem (CLT), when we repeatedly sample from a population and calculate the sample means, the distribution of those means will tend to become symmetric and bell-shaped, regardless of the shape of the original population distribution.
- Since the true population mean (parameter) is $30, the sample means would tend to cluster around this value, making it the center of the distribution.
c) How many dots will be on the dot plot described in (b)?
- 1000 dots.
Explanation:
- Since we are taking 1000 samples, each sample will contribute a dot to the plot, representing the sample mean. Therefore, after 1000 samples, the dot plot will contain 1000 dots.
d) What will each dot represent?
Each dot on the dot plot will represent the mean copayment for a month’s supply of topical painkiller ointment calculated from one sample of 75 regular users.
Explanation:
- In a sampling process, we take multiple random samples from the population, and for each sample, we calculate the mean of the copayments.
- The result of each sample is a single data point—one dot on the dot plot.
- Each of these dots represents the average copayment for a group of 75 users, not the individual payments, but rather the mean payment for that specific sample.
- When we plot these 1000 sample means, the result should form a distribution that reflects the underlying population’s average, and the shape of this distribution will be influenced by the Central Limit Theorem.
- Because the sample size is large (75 users per sample), the sample means should be distributed symmetrically and closely clustered around the population mean of $30.
This process of sampling and plotting is important in statistics because it allows us to estimate population parameters (like the true mean of $30) based on sample data. By analyzing the distribution of sample means, we can understand the variability and accuracy of our estimates.