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9.17 (Example 5) Age of Used Vans The mean age of all 118 used Toyota vans for sale (see exercise \(9.16\) ) was \(3.1\) years with a standard deviation of \(2.7\) years. The distribution of ages is rightskewed. For a statistics project, a student randomly selects 35 vans from this data set and finds the mean of the sample is \(2.7\) years with a standard deviation of \(2.1\) years. a. Find each of these values: \(\mu=? \quad \sigma=? \bar{x}=? \quad s=?\) b. Which of the values listed in part a are parameters? Which are statistics? c. Are the conditions for using the CLT fulfilled? What would be the shape of the approximate sampling distribution of a large number of means, each from a sample of 35 vans?

Step by step solution

01

Assign the Given Values

From the question, it's given that the mean age of all vans (\(\mu\)) is 3.1 years and the standard deviation (\(\sigma\)) is 2.7 years. For the sample of the 35 vans, the mean (\(\bar{x}\)) is 2.7 years and the standard deviation (s) is 2.1 years.

02

Differentiate between Parameters and Statistics

The parameters are numerical values that describe the characteristics of a population, in this case \(\mu = 3.1\) and \(\sigma = 2.7\) are parameters. Statistics are numerical values that describe the characteristics of a sample, so here \(\bar{x} = 2.7\) and \(s = 2.1\) are Statistics.

03

Check the Conditions for CLT

The conditions for applying the Central Limit Theorem (CLT) are as follows: The sample must be random, the sample size must be large enough (typically n>30), and the sampling should be done with replacement or the population is at least 20 times larger than the sample. In this case, all conditions are fulfilled because the vans were selected randomly, there are 35 vans in the sample (n>30), and the population size (118) is more than 20 times the sample size.

04

Predict the Shape of the Sampling Distribution

By the CLT, when conditions are satisfied, the sampling distribution of the sample means would be approximately normally distributed. However, since the original distribution is right-skewed, even with a reasonable sample size, it may still keep some degree of skew. Therefore, the distribution would likely be somewhat right-skewed, but less so than the original distribution.

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Key Concepts

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

Sampling Distribution

In statistics, a sampling distribution refers to the probability distribution of a given statistic based on a random sample. It quantifies how a statistic, like the sample mean, behaves across different samples. In our exercise, we examine the distribution of mean ages of randomly chosen used Toyota vans.

Consider the following details about sampling distributions:

• The mean of the sampling distribution (\( \mu_{x} \)) is equal to the population mean (\( \mu \)).
  
• As the sample size increases, the sampling distribution becomes less spread out, i.e., it has lower variance.
  
• The variance of the sampling distribution gets smaller with larger samples and is calculated as \( \sigma^2/n \) where \( \sigma \) is the population standard deviation and \( n \) is the sample size.
  

The sampling distribution helps answer questions about the population based on the sample, providing a bridge from the data we have to the data we want to know.

Parameters vs Statistics

In statistics, it is important to differentiate between parameters and statistics. Parameters are numerical values that summarize data for an entire population, whereas statistics summarize data from a sample. To put it in context:

• Parameters:
  • Often denoted with Greek letters like \( \mu \) and \( \sigma \).
    
  • For our exercise, \( \mu = 3.1 \) years and \( \sigma = 2.7 \) years are population parameters.
    
• Statistics:
  • Typically represented using Roman letters like \( \bar{x} \) and \( s \).
    
  • Thus, \( \bar{x} = 2.7 \) years and \( s = 2.1 \) years are sample statistics in our example.
    

Understanding this difference helps in analyzing the extent to which a sample represents its population. It is fundamental in making precise inferences and predictions.

Normal Distribution Approximation

The Central Limit Theorem (CLT) is pivotal in understanding why sampling distributions of the sample mean tend to be normal. According to CLT, regardless of the population distribution's shape, if the sample size is sufficiently large (usually more than 30), the sampling distribution of the sample mean will approximate a normal distribution.

Here's how it applies to our exercise:

• Since the sample size is 35, which is larger than 30, CLT suggests the distribution of sample means will be approximately normal.
  
• Although the original distribution of van ages is right-skewed, the large sample size helps mitigate this skewness.
  
• In practice, the distribution may still have a slight skew, reflecting the population's traits, but will generally conform to near-normality.
  

The normal distribution approximation simplifies statistical analysis by providing a reliable model for inference, an essential tool in statistics.