Alicia would like to know if there is a difference in the average price between two brands of shoes. She selected and analyzed a random sample of 40 different types of Brand A shoes and 33 different types of Brand B shoes. Alicia observes that the boxplot of the sample of Brand A shoe prices shows two outliers. Alicia wants to construct a confidence interval to estimate the difference in population means. Is the sampling distribution of the difference in sample means approximately normal

The sampling distribution of the difference in sample means approximately normal.

Given data:

Number of variants for Brand A shoes is, 40.

Number of variants for Brand B shoes is, 30.

The problem is based on the Central Limit Theorem. As per the Central Limit Theorem,  if we have a population with mean μ and standard deviation σ and appropriately huge random samples (n ≥ 30) are selected from the population with replacement, then the distribution of the sample mean will be approximately normally distributed.

Since, the sampling distribution of the sample mean price of both the brands is normal then the sampling distribution of the difference between the prices will also be normal.

Thus, we can conclude that the sampling distribution of the difference in sample means approximately normal.

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The sampling distribution of the difference in sample means approximately normal.

Step-by-step explanation:

The Central Limit Theorem states that if we have a population with mean μ and standard deviation σ and appropriately huge random samples (n ≥ 30) are selected from the population with replacement, then the distribution of the sample mean will be approximately normally distributed.

In this case Alicia selected a random sample of 40 different types of Brand A shoes and 33 different types of Brand B shoes.

Both the sample sizes are quite large.

So, the central limit theorem can be used to approximate the distribution of the sample mean price of shoes of brand type A and B.

Since, the sampling distribution of the sample mean price of both the brands is normal then the sampling distribution of the difference between the prices will also be normal.

Thus, the sampling distribution of the difference in sample means approximately normal.

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