Question

A) state the complete Central Limit Theorem (CLT)

B) explain why we need the theoretical idea of sampling distributions in a hypothesis test even though we only take one sample to decide between the hypothesis.

C) relate each part of the formula r= X-Mean₀ / S/ square root of n

D) Explain what type 1 and type 2 errors are

E) explain how it is possible to conduct the correct test flawlessly using a simple random sample of sufficient size and still commit a Type 1 and type 2 error. Use the idea of a sampling distribution in your explanation.

Answer 1

SOLUTION

a)

The Central Limit Theorem states that the sampling distribution of the sample means approaches a normal distribution as the sample size gets larger — no matter what the shape of the population distribution.

This fact holds especially true for sample sizes over 30.

All this is saying is that as you take more samples, especially large ones,

your graph of the sample means will look more like a normal distribution.

b)

Researchers often use a sample to draw inferences about the population that sample is from.

To do that, they make use of a probability distribution that is very important in the world of statistics:

the sampling distribution. It is theoretical distribution.

The distribution of sample statistics is called sampling distribution.

sampling distribution(theoretical distribution) is the probability distribution of a given statistic based on a random sample.

Sampling distributions are important in statistics because they provide a major simplification to statistical inference like in hypothesis testing for regression coefficients.

It allows analytical considerations to be based on the sampling distribution of a statistic, rather than on the joint probability distribution of all the individual sample values.

One can test the significance of regression coefficients from a sample drawn and hypothesis testing

Hypothesis testing is sometimes called confirmatory data analysis, in contrast to exploratory data analysis.

In frequency probability, these decisions are almost always made using null-hypothesis tests (i.e., tests that answer the question

Assuming that the null hypothesis is true, what is the probability of observing a value for the test statistic that is at least as extreme as the value that was actually observed)

One use of hypothesis testing is deciding whether experimental results contain enough information to cast doubt on conventional wisdom.

c)

According to CLT, if n is sufficiently large (<30 )

We will use the t distribution

Where x bar=Sample mean

Population mean

s=sample standard deviation

n=sample size

Frequently we interested in μ and estimate it using , so we need to know about the sampling distribution .

Theory says that for random samples of size n from any population

Frequently we interested in μ and estimate it using , so we need to know about the sampling distribution .

Theory says that for random samples of size n from any population

Mu-X-bar is equal to Mu. Mu stands for the population mean, and Mu-X-bar stands for the mean of the sampling distribution of all the sample mean.

The standard deviation of the sampling distribution is symbolized by Sigma-X-bar and is equal to Sigma divided by the square root of n.

The X-bar is added to make clear that we are talking about the standard deviation of the sampling distribution in which the scores are sample means, or in other words, X-bars.

Sigma stands for the standard deviation in the population. And n stands for the sample size.

d)

Type I error is the rejection of a true null hypothesis (also known as a "false positive" finding or conclusion

Type II error is the failure of rejecting a false null hypothesis also known as a "false negative" finding or conclusion