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Chapter 7: Problem 65

Each student should bring 10 coins to class. For each coin, observe its age, the difference between the current year and the year on the coin. a. Using all the students' observations, the class should construct a histogram of the sample ages. What is its shape? b. Now each student should find the mean for that student's 10 coins, and the class should plot the means of all the students. What type of distribution is this, and how does it compare to the one in part a? What concepts does this exercise illustrate?

Short Answer

Expert verified
a. The histogram of coin ages is generally skewed. b. The distribution of means is approximately normal. This showcases the Central Limit Theorem.

Step by step solution

01

Understand the Histogram of Coin Ages

Each student brings 10 coins and notes the age of each by subtracting the year on the coin from the current year. The class then plots these ages into a histogram. The shape of the histogram would typically be right-skewed because older coins are less common. "Current" years like 2023 would have more coins, and older years would be rare.
02

Calculate Individual Student Mean Ages

Each student calculates the mean age of their 10 coins. This provides an average age for each set of coins that the students have. These mean values offer insight into the general trend of coin ages from the perspective of each student's sample.
03

Plot the Distribution of Means

Plot all the students' calculated means to visualize the distribution. Often, this plot resembles a normal distribution, reflecting the Central Limit Theorem. Even if the original distribution of ages was skewed, the means converge to a normal distribution when many samples are taken.
04

Analyze and Compare Distributions

The histogram from Step 1 likely is right-skewed, showing raw, individual age data with a potential for many young coin ages clustering at the low end. In Step 3, the distribution of means, as per Central Limit Theorem, tends to be more bell-shaped and normal. This transformation and summarization illustrate how sample means form a normal distribution, given a sufficient number of samples.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Histograms
A histogram is a graphical representation of data using bars of different heights. Each bar groups numbers into ranges. In this exercise, students are encouraged to plot the ages of their coins. Each coin's age is defined as the current year minus the year marked on the coin. The histogram is then plotted with the number of coins on the y-axis and age ranges on the x-axis.
  • Histograms are useful for visualizing the distribution of a dataset.
  • They help in identifying the shape of the data distribution, such as whether it's symmetric, skewed, or has any outliers.
  • The height of each bar represents the number of data points within a particular range.
By observing the histogram constructed from the coin ages, you can often determine the distribution's overall shape, which is commonly right-skewed for coin age due to the abundance of more recent coins.
Computing the Sample Mean
In statistics, the sample mean is an average of a set of observations. Each student consolidates their 10 coins and computes the mean age. It's done by summing all ages and dividing by the number of coins:
designated as:\[\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i\]
where \( n \) is the number of observations and \( x_i \) represents each individual coin age.
  • Sample mean gives a summary statistic that represents the average value of the ages collected.
  • Individually, each student has a similar statistical viewpoint of coin age trends.
  • Despite variations, the sample mean becomes a reliable measure of central tendency for each student鈥檚 set of coins.
The calculated sample means contribute to understanding larger trends when considering all students' data as a whole.
Characteristics of Right-Skewed Distribution
A right-skewed distribution, often called positively skewed, features a longer tail on the right side. In the exercise featuring coins, this likely occurs because recent coins are more frequently collected than older ones, making higher ages more unusual.
  • Most data points are concentrated at the lower ages, such as new coins.
  • The mean is usually higher than the median due to the influence of the few older coins in the dataset.
  • Analyzing skewness helps us understand the spread and tendency of data.
Recognizing this right-skew in the initial histogram aids in predicting how the dataset behaves before considering the influence of the Central Limit Theorem through sample means.
Normal Distribution and the Central Limit Theorem
Normal distribution, or the bell curve, commonly appears in datasets with adequate sample sizes due to the Central Limit Theorem (CLT). When a large number of samples are taken and their means are plotted, the result is often a normal distribution.
The Central Limit Theorem states:
  • For sufficiently large sample sizes, the sampling distribution of the sample mean will approximate a normal distribution.
  • This emergence of normality occurs regardless of the initial data distribution.
  • The impact is clearer as the sample size becomes larger, emphasizing the importance of repeated sampling.
In the student exercise, despite the initial right-skew of individual coin ages, the distribution of student-calculated means aligns more closely to a normal distribution, illustrating the power of the Central Limit Theorem in statistical analysis.

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Most popular questions from this chapter

How would you explain to someone who has never studied statistics what a sampling distribution is? Explain by using the example of polls of 1000 Canadians for estimating the proportion who think the prime minister is doing a good job.

In one lottery option in Canada (Source: Lottery Canada), you bet on a six- digit number between 000000 and \(999999 .\) For a \(\$ 1\) bet, you win \(\$ 100,000\) if you are correct. The mean and standard deviation of the probability distribution for the lottery winnings are \(\mu=0.10\) (that is, 10 cents) and \(\sigma=100.00\). Joe figures that if he plays enough times every day, eventually he will strike it rich, by the law of large numbers. Over the course of several years, he plays 1 million times. Let \(\bar{x}\) denote his average winnings. a. Find the mean and standard deviation of the sampling distribution of \(\bar{x}\). b. About how likely is it that Joe's average winnings exceed \(\$ 1,\) the amount he paid to play each time? Use the central limit theorem to find an approximate answer.

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In \(2005,\) a study was conducted in West Texas A\&M University. It showed that the average student debt in the United States was \(\$ 18,367\) with a standard deviation of \$4709. (Source: http://swer.wtamu. edu/sites/default/files/Data/15-26-49-178-1-PB.pdf.) a. Suppose 100 students had been randomly sampled instead of collecting data for all of them. Describe the mean, standard deviation, and shape of the sampling distribution of the sample mean. b. Using this sampling distribution, find the \(z\) -score for a sample mean of \(\$ 20,000\) c. Using parts a and b, find the probability that the sample mean would fall within approximately \(\$ 1000\) of the population mean.

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