Consider the following hypothesis test: H0 : μ=100 Ha : μ≠ 100 A sample of 65 is used.

Consider the following hypothesis test: H0 : μ=100 ...

Key Concepts

P-value

The p-value is the probability, under the assumption that the null hypothesis is true, of obtaining a test statistic at least as extreme as the one computed from the sample data. It provides a measure of the evidence against the null hypothesis: a very small p-value indicates strong evidence in favor of the alternative hypothesis, while a large p-value suggests that the observed data are consistent with the null hypothesis.

Hypothesis Testing

Hypothesis testing is a statistical method that uses sample data to evaluate a claim about a population parameter. It involves setting up a null hypothesis, which represents a default position or no effect, and an alternative hypothesis, which represents a competing claim. The goal is to determine whether there is enough evidence in the sample to conclude that the alternative hypothesis is more likely to be true than the null hypothesis.

Test Statistic

The test statistic is a standardized value computed from the sample data that is used to decide whether to reject the null hypothesis. In the context of mean testing with unknown population standard deviation, it is typically computed using the t-distribution. The test statistic quantifies the distance between the sample mean and the hypothesized population mean in terms of the standard error of the mean.

Two-Tailed Test

A two-tailed test is used when the alternative hypothesis involves the possibility of the parameter being either greater than or less than the value specified in the null hypothesis. In this type of test, extreme results are considered significant in both directions, and the p-value is calculated by taking the sum of the probabilities in both tails of the test statistic’s distribution.

Significance Level

The significance level, denoted by alpha (?), is a threshold set by the researcher to determine when to reject the null hypothesis. It represents the maximum probability of making a Type I error, which is rejecting a true null hypothesis. A commonly used significance level is 0.05, meaning that there is a 5% risk of incorrectly rejecting the null hypothesis if it is actually true.

Transcript

00:01 In this problem, we're given the following, we have to find a p value and compare it to our alpha of 0 .05, given that our null hypothesis is that mu is equal to 100.

00:09 Our alternative hypothesis is that mu is not equal to 100, and we have a sample size of 65.

00:13 We have to do this for three different situations.

00:15 So in the first situation, we are given that our sample mean is equal to 103, and our sample standard deviation is equal to 11 .5.

00:29 So because we have a sample standard deviation, we have to use the t test.

00:34 So we have to find a t test statistic.

00:36 This is a formula for a t test statistic that is equal to the sample mean minus the population mean over the sample standard deviation divided by the square root of n.

00:46 So if we plug the values in, that is equal to 103.

00:51 Change colors for this.

00:52 103 minus 100 over 11 .5 divided by our sample size of 60, the square root of our sample size of 60.

01:03 And we get that our t test statistic equal to 2 .09.

01:09 So when we draw a t distribution, we get that t equals 0 is here.

01:16 What we are finding is the area to the right and under the curve of t equaling 2 .09.

01:23 So this area represents the probability that t is greater than or equal to 2 .09.

01:31 And how do we do that? we go to our t chart, if that's the name for it, and we find the degrees of freedom that we have.

01:44 So to calculate the degrees of freedom, we take our sample size and subtract one from it.

01:50 So we have 64 degrees of freedom because 65 minus 1 is 64.

01:55 And we scan in the degrees of freedom column, and we find that we have 64 degrees of freedom.

02:01 So we're going to work on this column.

02:04 We're going to work in this column over here.

02:07 And we scan across this row and see where our t test statistic lies.

02:13 So it lies between 1 .998 and 2 .836.

02:17 When translated to a p value that is in between 0 .025 and 0 .01.

02:23 So the probability that t is greater than 2 .09, probability that t is greater than 2 .09, probability that t is greater than 2 .0 .3 .2 .2.

02:35 0 .09 lies between what were those values 0 .025 and 0 .01.

02:56 Oh, the other way.

02:58 0 .01, 0 .025 and 0 .01.

03:06 I made the same mistake that i did originally.

03:08 Okay.

03:09 So this is the probability that t is greater than are equal to 2 .09.

03:14 However, because our alternative hypothesis is that mu is not equal to something, we have a two -tailed.

03:21 T test.

03:24 So if we go back here we have to multiply this value by two to find the p value.

03:31 So 0 .02 is less than or equal to two times p t greater than or equal to 2 .09 which is less than or equal to 0 .05 to find our p value.

03:47 So this right here is the range of our p.

03:54 Okay? okay.

03:56 And now we are going to compare our p value to our alpha of 0 .05.

04:00 So 0 .02, which is less than or equal to p, which is less than or equal to 0 .05, is less than or equal to our alpha of 0 .05.

04:12 So we can reject the null hypothesis.

04:17 All right.

04:18 Now we're going to do the same thing for the next problem.

04:21 The next part, where we have a sample mean of 96 .5 and a sample start standard deviation of 11 .0.

04:28 So if we use the formula for a t test statistic right here, it's the sample mean by the population mean, minus the population mean divided by the sample standard deviation divided by the square root of n, we get 96 .5 our sample mean minus 100 our population mean over our sample standard deviation of 11 .0 divided by the square root of our n, which is 65.

04:54 And we get a value of negative 2 .55...

Please give Ace some feedback