A worker for the American Red Cross wants to predict the average number of blood donations of college students. He takes a random sample of 900 students. But then he is puzzled and asks you:
" In my sample most of the students have zero donations and the percentages go down as the number of donations goes up. It does not sound look anything like a Normal distribution, so how can I use a Normal distribution in making my prediction?"
To understand why a normal distribution may still be useful in making predictions, let's explore the concept of the Central Limit Theorem (CLT). The CLT states that the sampling distribution of the mean of a sufficiently large number of independent and identically distributed random variables will be approximately normally distributed, regardless of the shape of the original distribution.
In your case, while the distribution of donations among college students may not follow a normal distribution, you can still use the normal distribution for prediction if certain conditions are met. These conditions are:
1. Independence: The observations within your sample should be independent, meaning the donation behavior of one student should not influence the behavior of another student.
2. Randomness: The sample should be selected randomly, ensuring a fair representation of the population and reducing potential biases.
3. Sufficiently large sample size: As a rule of thumb, a sample size greater than 30 is often considered to be "sufficiently large" for the CLT to apply. In your case, having a sample size of 900 students satisfies this condition.
By relying on the CLT, you can assume that the sampling distribution of the mean of the donation numbers (i.e., the average number of donations) will be approximately normally distributed. This allows you to make predictions using the properties and techniques associated with the normal distribution.
Keep in mind that while a normal distribution may be useful for making predictions about the average number of blood donations, it may not accurately capture the distribution of individual donations. In such cases, other statistical techniques or approaches might be more appropriate.