Confidence Interval

Section 5.1 - Two Normal Means

Case 2 (Unknown population variance)

Note - The pooled estimator S-squared p is based on the maximum likelihood estimator but adjusted to give an unbiased estimator.

What does it mean when they it is based on the MLE and what adjustment was made?

Thanks.

 

Query No. 2

HYPOTHESIS TESTING

I want some clarification on the basic concept of hypothesis testing.

From what I understand, SIGNIFICANCE LEVEL is the probability of committing a TYPE I ERROR i.e. it is the probability of rejecting the null hypothesis even if it is true (correct me if I am wrong).

Why do we reject the null hypothesis when the P-value is less than the significance level?

Said that, what exactly is P-value? "Suppose the test statistic is equal to Z. The P-value is the probability of observing a test statistic as extreme as Z, assuming the null hypothesis is true." Can someone simplify this sentence for me?

I would love an example. Thanks a lot.

 

Suppose our null hypothesis is that the mean age of students = 22 and our alternative hypothesis is that the mean age is not equal to 22.

Suppose we take a sample and find that the sample mean age is 26.

We can deal with this in two main ways:

(1) What is the probability of getting this sample result if Ho is true? This is called the probability method and the probability is called the p-value.

(2) What value of the sample mean would convince me that Ho is true/false? This is called the critical value method.

Using (1), we can calculate the probabilty of getting a sample mean as extreme as this on both sides (ie >26 or <18) if the population mean is actually 22. Suppose this probability turns out to be large, eg 34%. We would say that getting a sample mean as extreme as this if the population mean is actually 22 is not very unusual at all so therefore we would say that we do not have sufficient evidence to reject Ho. However, if the probability of getting a sample mean as extreme as this if the population mean is actually 22 is very low, eg 3%, we would say that this is very unusual and therefore we would reject Ho.

We usually use 5% as a cut-off point, ie we usually run a 5% Type I error - so there is a 5% chance of rejecting Ho when it is in fact true. This is called the significance level of the test.

Using (2) we can determine the critical values of the sample mean, C1 and C2 (or the test statistic, eg Z1 and Z2) such that the probability of a result lower than C1 or higher than C2 is 5%. Then if our sample mean (or the test statistic) is in this critical range, we will reject Ho at the 5% level.

 

Thanks a lot freddie. I started looking at examples and got a good idea about hypothesis. But could you also throw some light on the Neyman Pearson Lemma?

Why do we calculate the ratio of the likelihoods?

What does that suggest?

Thanks again.

 

I am still awaiting an answer. Can someone please provide me a logical explanation of the Neyman Pearson Lemma?

There are not many problems related to that in the study material (Actually only one). I am still not sure how does it work? I mean the Best Critical Region, The ratio of the likelihoods?

I would be grateful if someone can take out some time and help me with it.

Thanks in advance

 

The Neyman-Pearson lemma provides the theoretical justification for the tests that we use.

It shows how, out of all the possible critical regions with a Type I error of alpha, we can determine a critical region that minimises beta, the probability of a Type II error. This will then be the most powerful test of size alpha.

We determine C such that:

• L(under Ho)/L(under H1) < k for all x values in C and
• L(under Ho)/L(under H1) > k for all x values outside C

where k is some constant chosen so that the Type I error is alpha.

This ratio of the likelihoods is measuring the plausibility of Ho relative to H1. If this ratio is sufficiently small (indicating that Ho is not very likely), we'll reject Ho.

 

Thanks again freddie.

Thanks John. I didnt know that. But since it was an entirely new concept for me I thought I should know that. Thanks for your input.