4 min read•january 2, 2023

B

Brianna Bukowski

Josh Argo

Jed Quiaoit

To find the standard deviation of differences in sample means, divide the variances by each sample size before square rooting to find the overall standard deviation. Just like with proportions, the **“Pythagorean Theorem of Statistics” **applies to sampling distributions for the difference in two means as well. Here are the formulas for the needed parameters for sampling distribution of difference of two means. 🕯️

When you are working with differences between sample means, you can use the sampling distribution of the differences to make inferences about the difference between the population means. 🙌

If the two population distributions can be modeled with a normal distribution, then the sampling distribution of the difference in sample means x̄1 - x̄2 can also be modeled with a normal distribution. This means that you can use statistical techniques that rely on normality, such as confidence intervals and hypothesis tests, to make inferences about the difference between the population means based on the sample data.

If the two population distributions cannot be modeled with a normal distribution, the sampling distribution of the difference in sample means x̄1 - x̄2 can still be approximately normal if both samples are **large enough**. This is due to the Central Limit Theorem, which states that the sampling distribution of the sample mean becomes approximately normal as the sample size increases, regardless of the shape of the population distribution. As a result, **if both samples are large enough (e.g., have sample sizes of at least 30)**, you can still use normal-based techniques to make inferences about the difference between the population means. 🎈

Suppose that you are a publisher trying to compare the sales of two different genres of books: romance novels and science fiction novels. You decide to use random samples of 50 romance novels and 50 science fiction novels from your inventory, and you collect data on the number of copies sold for each book. After analyzing the data, you find that the sample mean number of copies sold for romance novels is 500 copies with a standard deviation of 100 copies, and the sample mean number of copies sold for science fiction novels is 400 copies with a standard deviation of 150 copies. 📚

On the other hand, if science fiction novel readers are more likely to buy books online or to be members of certain online communities, the sample of science fiction novels could be biased toward lower levels of sales and produce an underestimate of the population mean. This could lead to an incorrect estimate of the difference in population means between the two genres of books.

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