Question

Collette is self-employed, selling items at home interior parties. she wants to estimate the average amount a client spends at each party. a random sample of 35 clients' receipts gave a sample mean of $34.70 and historically we know the population standard deviation of spending at these parties to be $4.85.

explain why you may assume your data is normally distributed.

Answer 1

The data can be assumed to be normally distributed due to the Central Limit Theorem.

Step-by-step explanation:

In this case, we can assume that the data is normally distributed because of the Central Limit Theorem. The Central Limit Theorem states that regardless of the shape of the population distribution, if the sample size is large enough (typically, n > 30), the sampling distribution of the sample mean will be approximately normal. Since the sample size of 35 is greater than 30, we can assume that the sampling distribution of the sample mean, which is the average amount a client spends at each party, follows a normal distribution.

Note that the population standard deviation is not used to determine whether the data is normally distributed. Instead, it is used to calculate the standard error of the sample mean, which is a measure of the variability of the sample means.

We may assume Collette's data on client spending is normally distributed due to the application of the Central Limit Theorem which states that for a sufficiently large sample size (typically n ≥ 30), the sample means will be approximately normally distributed, regardless of the population's distribution. Since Collette has a sample size of 35, which is greater than 30, and since the population standard deviation is known, we can safely assume normality for the purpose of estimation and statistical testing.