Two boats, the Prada (Italy) and the Oracle (U.S.A.), are...

Name: Problem 38, Chapter 11: Basic Statistics for Business and Economics
Uploaded: 2022-04-02T15:26:27-08:00
Duration: 4 min 26 s
Channel: Sheryl Ezze
Description: Two boats, the Prada (Italy) and the Oracle (U.S.A.), are competing for a spot in the upcoming America's Cup race. They race over a part of the course several times. Below are the sample times in minutes. At the .05 significance level, can we conclude that there is a difference in their mean times?

Two boats, the Prada (Italy) and the Oracle (U.S.A.), are competing for a spot in the upcoming America's Cup race. They race over a part of the course several times. Below are the sample times in minutes. At the .05 significance level, can we conclude that there is a difference in their mean times?

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions or inferences about population parameters based on sample data. It involves formulating a null hypothesis, which represents a statement of no effect or no difference, and an alternative hypothesis, which suggests that there is an effect or difference, and then using test statistics to decide which hypothesis is more supported by the data.

Null and Alternative Hypotheses

The concepts of the null and alternative hypotheses are central to statistical hypothesis testing. The null hypothesis (H?) typically posits that there is no difference or effect—in this case, that the mean times of the two boats are equal—while the alternative hypothesis (H?) claims that there is a difference between the two means. This framework allows researchers to objectively assess the evidence provided by the data.

Two-Sample t-Test

The two-sample t-test is a statistical method used to determine whether there is a significant difference between the means of two independent groups. It is particularly useful when the sample sizes are small and the population variances are unknown. By comparing the computed t statistic to a critical value from the t-distribution, one can decide whether to reject the null hypothesis of equal means.

Significance Level

The significance level, often denoted by alpha (?), is a threshold used to determine the probability of rejecting the null hypothesis when it is actually true. At a common significance level of 0.05, there is a 5% risk of committing a Type I error, which is the error of incorrectly concluding that there is a difference between the groups when no such difference exists.

Transcript

0:00 So we have two boats.

00:01 We have the first boat, which is from italy, and the second boat, which is from the u .s.

00:06 And we're going to assume that their mean times on this course is the same, and alternately that the mean times are different.

00:14 And we're to do this at a 5 % significance level.

00:18 And so we have for the first group, the italian group, they had a sample size of their runs of 10 versus the u .s .a.

00:28 Had 12.

00:28 And the mean time from the italian group was 12 .17 minutes and a sample standard deviation of 1 .056 minutes.

00:43 And for the u .s.

00:46 Group, the mean was 14 .875 and a sample standard deviation of 2 .208 minutes.

00:54 Now, we are going to be assuming in this technique that these two standard deviations, are equal.

01:01 There is another technique for doing this without assuming they're equal, however your book hasn't shown you that.

01:07 So we will be assuming they're equal and that causes us to have to find our pooled standard deviation or our pooled variance.

01:14 And that is found by taking one less than the sample size times the variance of the first group and then one less than the sample size of our second group times the variance of our second group.

01:28 And then we divide that by the number of degrees of freedom.

01:31 Which is going to be the sum of the two populations, or the sum of the two sample sizes less two.

01:36 So we can see that our degrees of freedom will be 20.

01:40 And let's finish finding what this is, and then we will find what our chart is for where we'll be rejecting the null, and we'll find the test statistic.

01:52 So we have left parentheses nine times 1 .056 squared, so we're kind of weighting these two variances.

02:01 And this one has a higher weight because there is this larger sample size.

02:06 And we have 11 times the 2 .208 squared, close off that parenthesis, and then we'll be dividing by 20.

02:14 And we get that that pooled variance is approximately this value.

02:23 And again, kind of look at it...

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