The null and alternate hypotheses are: H0: μ1=μ2 H1: μ1≠μ2 A random sample of 15 obser

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

The null and alternate hypotheses are: $$\begin{array}{l}H_{0}: \mu_{1}=\mu_{2} \\H_{1}: \mu_{1} \neq \mu_{2}\end{array}$$ A random sample of 15 observations from the first population revealed a sample mean of 350 and a sample standard deviation of $12 .$ A random sample of 17 observations from the second population revealed a sample mean of 342 and a sample standard deviation of $15 .$ At the .10 significance level, is there a difference in the population means?

The null and alternate hypotheses are:
$$\begin{array}{l}H_{0}: \mu_{1}=\mu_{2} \\H_{1}: \mu_{1} \neq \mu_{2}\end{array}$$
A random sample of 15 observations from the first population revealed a sample mean of 350 and a sample standard deviation of $12 .$ A random sample of 17 observations from the second population revealed a sample mean of 342 and a sample standard deviation of $15 .$ At the .10 significance level, is there a difference in the population means?

Key Concepts

Test Statistic and Degrees of Freedom

The t-test statistic is computed to quantify the difference between group means in relation to the variability observed in the samples. The degrees of freedom, which are determined by the sample sizes (and sometimes adjusted when variances are assumed unequal), are used to reference the correct distribution when determining statistical significance.

Significance Level

This is a threshold probability, defined before the test, which is used to determine whether the observed data is sufficiently unlikely under the null hypothesis to warrant rejection of the null. A significance level of 0.10 indicates a 10% risk of committing a Type I error.

Null and Alternative Hypotheses

These are statements set up in hypothesis testing to compare parameters of interest. The null hypothesis typically represents no effect or no difference, while the alternative hypothesis represents the existence of an effect or a difference. In comparing two population means, the null hypothesis often states that the means are equal, and the alternative suggests that they are not.

Independent Samples t-Test

This is a statistical method used to compare the means of two independent groups based on sample data. It involves calculating a t-statistic from the sample means, sample standard deviations, and sample sizes to assess if any observed difference is statistically significant.

Transcript

00:01 So we're again assuming that the two groups have means that are equal and alternately the means are different.

00:09 And we have our first sample was only 15 large, and that revealed a mean of 350 with a standard deviation of 12.

00:21 And our second sample size is 17 and had a mean value of 342.

00:30 So they look different, but are they significantly? different.

00:33 And the second sample standard deviation was 15.

00:37 Now these are sufficiently small that's going to cause us to have to use a t statistic.

00:43 And also we're going to assume that these two come from two independent populations where the variances or the standard deviations are equal.

00:51 So we are going to need to find that pooled p.

00:53 And let's find what that pooled, pooled variance is, i should say.

00:57 And that pooled variance, we need to take one less than this sample size, so 14 times the 12 squared, which is 144, plus, and we'll take that one less than the 17, which is 16 times that square of that 15, which is 225, and then we'll divide that by the number of degrees of freedom, and that is going to be that 15 plus 17 less 2.

01:24 So that's going to end up being 30.

01:25 So our degrees of freedom for our study is going to end up being 30.

01:31 And before i do that calculation, let me just quick draw a picture of what our, let's do it in blue, what our critical values are going to be.

01:40 So we're doing a two -tail test and we're using a 10 % significance level.

01:45 Therefore, we're going to put 0 .05 in each tail.

01:48 And we need to have 29 degrees of freedom and my table goes up that high.

01:53 So we want a t value with 29 degrees of freedom that has 5 % in that upper tail.

01:59 And 5 % in the upper tail corresponds with a 1 .699.

02:05 And so this lower one will be the opposite.

02:10 And we will reject if we are in either of those two regions, reject the null...

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