Identify the null hypothesis, alternative hypothesis, test...

Identify the null hypothesis, alternative hypothesis, test statistic, Pvalue or critical value(s), conclusion about the null hypothesis, and final conclusion that addresses the original claim. In a recent New York City marathon, 25,221 men finished and 253 dropped out. Also, 12,883 women finished and 163 dropped out (based on data from the New York Times). We want to use a 0.01 significance level to test the claim that the rate of those who finish is the same for men and women. a. Test the claim using a hypothesis test. b. Test the claim by constructing an appropriate confidence interval. c. Based on the results, do men and women finish the New York City marathon at the same rate?

Identify the null hypothesis, alternative hypothesis, test statistic, Pvalue or critical value(s), conclusion about the null hypothesis, and final conclusion that addresses the original claim.
In a recent New York City marathon, 25,221 men finished and 253 dropped out. Also, 12,883 women finished and 163 dropped out (based on data from the New York Times). We want to use a 0.01 significance level to test the claim that the rate of those who finish is the
same for men and women.
a. Test the claim using a hypothesis test.
b. Test the claim by constructing an appropriate confidence interval.
c. Based on the results, do men and women finish the New York City marathon at the same rate?

Key Concepts

Confidence Interval

A confidence interval, typically calculated for the difference between two proportions, provides a range of plausible values for this difference based on the sample data. This interval offers an estimate of the effect size and is used in part (b) of the analysis to check if zero (indicating no difference) lies within the interval, which informs the conclusion regarding the claim.

Statistical Conclusion

The statistical conclusion involves deciding whether to reject the null hypothesis based on the test statistic and its corresponding P-value (or by comparing the confidence interval against zero). It is a synthesis of all inferential evidence that either supports or refutes the claim that men and women finish the marathon at the same rate.

P-value and Critical Value

The P-value represents the probability of observing a test statistic as extreme as the sample one, assuming that the null hypothesis is true. Alternatively, critical values are specific thresholds determined by the chosen significance level (in this case, 0.01) that define the rejection region for the test. Both concepts are used to decide whether to reject or fail to reject the null hypothesis in hypothesis testing.

Null Hypothesis

The null hypothesis is a statement of no difference, specifying that the completion rates for both groups are the same. In inferential statistics, it serves as the default or starting assumption about the parameters of interest—in this case, that the probability of finishing the marathon is equal for men and women.

Alternative Hypothesis

The alternative hypothesis is a statement that there is a difference between the groups, suggesting that the finishing rates for men and women are not equal. This concept sets up the need to prove that the observed differences in sample data reflect a real difference in the population, rather than random variation.

Test Statistic

The test statistic is a numerical value calculated from sample data that measures the degree of divergence from what the null hypothesis would predict. In comparing two proportions, this is often computed using a z-test formula, which standardizes the difference between the groups relative to its expected variability under the null hypothesis.

Transcript

00:01 We're looking at finishing a race, and we're going to assume that the proportion of males and the proportion of females finishing a race are consistent, are equal.

00:12 And alternately, that the proportion of males is different from the proportion of females in finishing a race.

00:20 And so we are really assuming that the difference between these two when we get that test statistic is zero, and we want to find.

00:30 Find out if we have evidence consistent.

00:31 But this is a two -tailed test.

00:34 So we will find p -value combining two areas together.

00:38 So one other thing to watch out for is that first one, you had 25 ,221 who finished, and then it told you how many didn't.

00:49 So we have to total those together to get this is how many people participated.

00:54 And same thing here.

00:56 We had 12 ,883, who finished and then they told us how many didn't.

01:02 So we had a total of 13 ,046 people.

01:06 And we could show finding the p hat and doing the calculation for the formula.

01:10 But let's just continue on.

01:11 Let's find the z value.

01:13 So i have this into my calculator and have it as a not equal to.

01:21 And when we do, we find out that this difference creates a z value or a test statistic that is 2 .3029.

01:30 So it's up here, you know, over two standard deviations higher.

01:35 So this area we want to find.

01:38 So we want to find the probability of if the distribution has a difference of zero.

01:43 How often will we get this type of a difference in proportions or worse, higher? and then we're going to want to double that because of it being a two -tail test.

01:55 So we're going to find this area as well.

01:58 And when we do, we get our p values, 0 .2 .1...

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