The null and alternate hypotheses are: H0: μ1=μ2 H1: μ1≠μ2 A random sample of 10 obser

The null and alternate hypotheses are: H0: μ1=μ2 ...

Key Concepts

Degrees of Freedom

Degrees of freedom refer to the number of independent values that can vary in an analysis without breaking any constraints, and they are essential in determining the appropriate t distribution when conducting a t-test. In the context of a two-sample t-test, the calculation of degrees of freedom, which is influenced by the sample sizes, impacts the critical values used for significance testing.

Significance Level

The significance level, commonly denoted by alpha (?), is a predetermined threshold used to evaluate the strength of the evidence against the null hypothesis. It defines the maximum probability of committing a Type I error, i.e., rejecting a true null hypothesis, with common values being 0.05 or 0.01.

Two-Sample t-Test

The two-sample t-test is used to compare the means of two independent groups when the population variances are unknown and the sample sizes are relatively small. This test relies on the t distribution for determining whether the difference between the sample means is statistically significant, considering any variability in the data.

Null and Alternative Hypotheses

The null and alternative hypotheses are central to statistical tests. The null hypothesis (H0) represents the assumption of no effect or no difference in the parameter of interest, while the alternative hypothesis (H1) suggests the presence of an effect or a difference. The outcome of the hypothesis test hinges on comparing data-derived measures to these pre-stated hypotheses.

Hypothesis Testing

Hypothesis testing is a statistical framework used to decide between competing claims about a population based on sample data. It involves setting up a null hypothesis that typically represents the status quo and an alternative hypothesis that represents a deviation or a new claim, then using test statistics to determine whether the observed data provide enough evidence to reject the null hypothesis.

Transcript

00:01 So we want to find out whether we have a difference between these two means.

00:04 So we're going to assume that they're equal and alternately that they are different.

00:10 And we have our sample sizes from our two groups.

00:13 Our first sample size is only 10, yielding a mean of 23 and a sample standard deviation of 4.

00:22 And our second sample has an even smaller sample size of 8.

00:26 The mean is 26.

00:28 And a sample standard deviation of five.

00:32 Now, we're going to be assuming, our test is at a 5 % significance level, and because our sample sizes are so small, and we don't have any indication that these are coming from two normal populations, we're going to have to use a t test.

00:48 And so far, we're dealing with the idea that assuming that the two population standard deviations are equal, so we're going to have to find that pooled standard deviation or that pooled variance.

01:00 And so that pooled variance is found by taking one less than this sample size, so nine times the variance of our first group, which is 16, plus we take our second size less one or its number of degrees of freedom times the variance, which that will be 25.

01:20 And then that will be divided by the sum of these two less two.

01:24 And so 10 plus 8 minus 2.

01:28 And this denominator is also our number of degrees of freedom.

01:31 So let's get what that total is.

01:33 And i have 9 times 16 plus 7 times 25, and then dividing that by, and that looks like that is 16.

01:43 And so we end up getting that that variance is 19 .9375.

01:50 Now, since our number of degrees of freedom will end up being, again, this n1 plus n2 less 2, which is what our denominator is here, which is going to end up being 16.

02:04 So if we're dealing with the significant situation in which we have 5 % significance level, and we're doing a two -tailed test, let's just quick draw that little picture of that sampling distribution, we're going to have to put both.

02:20 Split that 5 % into the two tails.

02:24 And so we want to find the t value that has 16 degrees of freedom with 0 .025 in that upper tail...

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