Using Central Limit Theorem to Estimate Sample Means in a Stats Class"

In summary, the students in an evening stats and methods class were asked to generate 64 random numbers using a spinner. The mean for the class was 27 with a standard deviation of 20. The question asks how many students would be expected to have a sample mean less than 23.25, using the formula \frac{\sqrt{n}(\overline{x} - \mu)}{\sigma} with n = 180.
  • #1
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Homework Statement



Each of 180 students in an evening stats and methods class is asked to generate 64 random numbers with a "spinner" that selects numbers from 1 to 50, and then compute the mean of the 64 numbers. The mean for the class as a whole is 27 with a standard deviation of 20. How many of the students would be expected to have their sample means less than 23.25?


Homework Equations





The Attempt at a Solution



Would I use this form of the CLT: [tex]\frac{\sqrt{n}(\overline{x} - \mu)}{\sigma}[/tex]
 
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  • #2
No one has responded, so I think I'll weigh in - yes, that's the statistic to use, with n = 180.
 
  • #3
That's what I thought; thanks.
 

1. What is the Central Limit Theorem?

The Central Limit Theorem is a statistical theory that states that the distribution of sample means from a population will approach a normal distribution as the sample size increases. This means that even if the population is not normally distributed, the distribution of sample means will be approximately normal.

2. Why is the Central Limit Theorem important in statistics?

The Central Limit Theorem is important because it allows us to make inferences about a population based on a sample. It provides a way to estimate the population mean and other parameters with a certain level of confidence, even if the population is not normally distributed.

3. How is the Central Limit Theorem used to estimate sample means?

In a statistics class, the Central Limit Theorem is used to estimate the population mean by taking multiple samples from the population and calculating the mean of each sample. These sample means can then be plotted to form a sampling distribution, which will approach a normal distribution as the sample size increases. The mean of this sampling distribution can then be used as an estimate of the population mean.

4. What are the assumptions of the Central Limit Theorem?

The Central Limit Theorem assumes that the sample is randomly selected from the population, the sample size is large enough (usually n > 30), and that the observations in the sample are independent of each other. If these assumptions are met, then the Central Limit Theorem can be used to estimate sample means.

5. Can the Central Limit Theorem be used with any population?

The Central Limit Theorem can be used with any population, as long as the sample size is large enough and the other assumptions are met. However, if the population is already normally distributed, there is no need to use the Central Limit Theorem as the sample mean will already be an accurate estimate of the population mean.

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