The authors of the paper "External Factors and the Incidence...

The authors of the paper "External Factors and the Incidence of Severe Trauma: Time, Date, Season and Moon" (Injury [2014]: S93-S99) classified admissions to hospitals in Germany according to season. They wondered if severe trauma injuries were more common in some seasons than others. For purposes of this exercise, assume that there were 1200 trauma cases in the sample and that the sample is representative of severe trauma injuries in Germany. The data in the accompanying table are consistent with summary quantities given in the paper. Do these data support the theory that the proportion of severe trauma cases is not the same for the four seasons? Test the relevant hypotheses using a significance level of 0.05 .

Key Concepts

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis represents the presumption of no effect or no difference—in this context, it claims that the distribution of severe trauma injuries is the same across all seasons. The alternative hypothesis, on the other hand, posits that there is a difference in the distribution, meaning that some seasons have significantly different proportions of trauma cases than others.

Chi-Square Goodness-of-Fit Test

The Chi-Square Goodness-of-Fit Test is a statistical method used to determine whether the observed frequencies in categories match expected frequencies based on a specific hypothesis. It compares the observed counts to the counts expected under the assumption of an equal distribution, helping to assess if the differences are due to random chance or if they are statistically significant.

Significance Level

The significance level, commonly set at 0.05 in many studies, is the pre-determined threshold for rejecting the null hypothesis. It defines the probability of making a Type I error by declaring a difference when none exists. If the observed p-value is below this threshold, it is considered statistically significant and suggests sufficient evidence to reject the null hypothesis.

Degrees of Freedom

Degrees of freedom refer to the number of independent comparisons that can be made in the data without violating any constraints, typically calculated as the number of categories minus one for the Chi-Square Goodness-of-Fit Test. They are crucial in determining the critical value from the chi-square distribution against which the test statistic is compared.

p-Value

The p-value is the probability of obtaining results as extreme as those observed, under the assumption that the null hypothesis is true. A p-value less than the designated significance level (0.05) indicates that the observed data are unlikely to have occurred by chance, thus providing evidence to support rejection of the null hypothesis.

Transcript

00:01 So they collected data on pairs of fridays in a month, the 6th and the 13th, and they're comparing those together to see about the number of people that are admitted to the hospital on those.

00:19 They're pairs of fridays of the month, so we can use a paired t -test for this.

00:27 Now for our null hypothesis, 6 is 1 and the 13th would be 2.

00:37 We're gonna do subtract those and if there's no difference between them, that would be that it was 0 and h -a more people are admitted on friday the 13th.

00:57 So if we have more people admitted on friday the 13th, that means that we're gonna get negative numbers.

01:04 It's important to have these sketched out so that you can draw your conclusion.

01:11 Well, my p -value is 0 .0211.

01:17 So since that p -value is smaller than 0 .05, we can reject the null hypothesis.

01:30 This means there is convincing evidence or enough evidence or whatever that more people are admitted to the hospital on friday the 13th.

02:10 Okay, because the p -value is smaller than 0 .05, we're gonna reject that they're the same and there's enough evidence to say that the mean number of people admitted to the hospitals on friday the 13th is larger than other days, other fridays.

02:29 And then c wants us to check out the conditions.

02:33 We need to ask ourselves, were the days, the pairs, picked randomly? we don't know.

03:00 So this kind of makes it on sketchy ground because it doesn't say anything about the pairs of dates being selected at random.

03:18 Yeah, so that's kind of sketchy...

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