Assume that we collect a large (n>30) simple random sample of annual incomes of adults in the United States. Because the sample is large, can we approximate the distribution of those incomes with a normal distribution ? Why or why not?
Yes, whenever you have a large sample you can approximate it by using the normal distribution.
To determine whether we can approximate the distribution of incomes with a normal distribution, we need to consider a few factors.
In general, the Central Limit Theorem (CLT) states that the distribution of sample means, for a sufficiently large sample size, will approximate a normal distribution regardless of the shape of the original population distribution. However, it is important to note that the CLT specifically applies to the distribution of sample means, not the original variable itself.
In the context of your question, we are not interested in the distribution of sample means but rather the distribution of individual incomes. While a large sample size (n > 30) does make the CLT more applicable to the sample means, it does not directly address the distribution of individual incomes.
The distribution of individual incomes in the United States is likely to be positively skewed (skewed to the right) because incomes tend to have a long tail on the higher end. A normal distribution, on the other hand, is symmetric and bell-shaped.
To determine if the distribution of incomes can be approximated by a normal distribution, it is recommended to examine the skewness and kurtosis of the data. These statistical measures can provide insights into the shape and deviation from normality of the distribution. If the skewness is significantly different from zero (positive or negative) or if the kurtosis deviates substantially from the value of three (the kurtosis of a normal distribution), then it is less appropriate to approximate the distribution with a normal distribution.
In summary, while the Central Limit Theorem allows us to approximate the distribution of sample means with a normal distribution with a large sample size, it does not necessarily imply that the distribution of individual incomes can be approximated by a normal distribution. Further analysis, such as examining skewness and kurtosis, should be conducted to assess the appropriateness of this assumption.