**Would it be unusual for the mean of a sample of 3 to be 115 or more? Why or why not?**

Yes, it would be unusual for the mean of a sample of 3 to be 115 or more because the population’s standard deviation is 15, so the mean of a selection of 3 would be expected to be within 15 standard deviations of the population mean of 100. The mean of the sample would be expected to be between 85 and 115. A mean of 115 or more would be outside this range, so it would be considered unusual.

To calculate the probability of obtaining a sample mean of 115 or more with a sample size of 3, we use the z-score

The formula is: z = (x̄ – μ) / (σ / √n)

x̄ is = sample mean which is 115

μ = population mean (100)

s is the sample standard deviation

n is the sample size which is 3

z = (115 – 100) / (15 / √3) = 1.7321

This corresponds to a probability very close to zero; hence it would be unusual to observe a sample mean of 115 or more with a sample size of 3

**What if the size for each sample was increased to 20? Would a sample mean of 115 or more be considered unusual? Why or why not?**

No, a sample mean of 115 or more would not be considered unusual if the size of each sample was increased to 20. This is because the Central Limit Theorem states that the distribution of sample means will approach a normal distribution as the sample size increases. Therefore, the probability of getting a sample mean of 115 or more would be much higher if the sample size was 20.

Sample Size = 20:

(115 – 100) / (15 / √20) = 4.4721

This corresponds to an area very close to 1.

As the sample size increases, the distribution of sample means it gets close to a normal distribution, which is the Central Limit Theorem.

**Why is the Central Limit Theorem used?**

The Central Limit Theorem is used because it allows us to make inferences about the population means based on the sample mean. It tells us that the sampling distribution of the mean will be normally distributed, regardless of the shape of the population distribution, as long as the sample size is large enough. This means we can use the normal distribution to calculate the probability of a sample mean being within a specific range.