Slow response times by paramedics, firefighters, and...

Slow response times by paramedics, firefighters, and policemen can have serious consequences for accident victims. In the case of life-threatening injuries, victims generally need medical attention within 8 minutes of the accident. Several cities have begun to monitor emergency response times. In one such city, the mean response time to all accidents involving life-threatening injuries last year was $\mu=6.7$ minutes. Emergency personnel arrived within 8 minutes after 78$\%$ of all calls involving life-threatening injuries last year. The city manager shares this information and encourages these first responders to “do better.” At the end of the year, the city manager selects an SRS of 400 calls involving life-threatening injuries and examines the response times. Awful accidents (a) State hypotheses for a significance test to determine whether the average response time has decreased. Be sure to define the parameter of interest. (b) Describe a Type I error and a Type II error in this setting, and explain the consequences of each. (c) Which is more serious in this setting: a Type I error or a Type II error? Justify your answer.

Slow response times by paramedics, firefighters, and policemen can have serious consequences for accident victims. In the case of life-threatening injuries, victims generally need medical attention within 8 minutes of the accident. Several cities have begun to monitor emergency response times. In one such city, the mean response time to all accidents involving life-threatening injuries last year was $\mu=6.7$ minutes. Emergency personnel arrived within 8 minutes after 78$\%$ of all calls involving life-threatening injuries last year. The city manager shares this information and encourages these first responders to “do better.” At the end of the year, the city manager selects an SRS of 400 calls involving life-threatening injuries and examines the response times. Awful accidents
(a) State hypotheses for a significance test to determine whether the average response time has
decreased. Be sure to define the parameter of interest.
(b) Describe a Type I error and a Type II error in this setting, and explain the consequences of each.
(c) Which is more serious in this setting: a Type I error or a Type II error? Justify your answer.

Key Concepts

Type II Error

A Type II error happens when the null hypothesis is not rejected when the alternative hypothesis is actually true. In this scenario, it would mean failing to recognize a real decrease in response times. This error might prevent the acknowledgement of improvements or the implementation of strategies that could further enhance emergency services, possibly delaying necessary enhancements in the system.

Type I Error

A Type I error occurs when the null hypothesis is incorrectly rejected when it is in fact true. In practical terms, this error would mean concluding that response times have decreased when in reality they have not. This could lead to unnecessary changes or policies based on a misinterpretation of the data, potentially affecting resource allocation or public trust in the emergency response system.

Parameter of Interest

The parameter of interest is the specific quantity or characteristic of the population that the study aims to estimate or test. In the context of response times, the parameter of interest is the true mean response time of emergency services, which is directly linked to performance evaluation. Clearly defining this parameter is essential as it underpins the formulation of the null and alternative hypotheses.

Null and Alternative Hypotheses

The null and alternative hypotheses are statements about the population parameter that are investigated during hypothesis testing. The null hypothesis (H0) typically states that there is no effect or no change, while the alternative hypothesis (Ha) states that there is an effect, such as a decrease in response times in this context. The decision to reject the null hypothesis depends on whether the observed data are sufficiently inconsistent with it under a predetermined level of significance.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about a population parameter by comparing sample data against a claim made about that parameter. It involves setting up a null hypothesis that represents a status quo or a default position, and an alternative hypothesis which represents a claim the researcher seeks evidence for. Results from sample data help decide whether there is sufficient evidence to reject the null hypothesis in favor of the alternative.

Transcript

00:01 Slow response time by paramedics, firefighters, and policemen can have some serious consequences for accidental victims.

00:09 We're told that in one city, the mean response time to all accidents involving life -threatening injuries last year was an average of 6 .7 minutes.

00:21 Now, the city manager shares the information and encourages the first responders to do better.

00:28 And so he runs an srs of four.

00:30 400 calls involving life -threatening injuries.

00:34 Now, for our hypothesis for a significance test to determine the average response time, let's define our parameter.

00:42 So mu is going to be the mean response time for all accidents involving life -threatening injuries in the city.

00:52 And in this case, our null hypothesis, h -null, is going to be that the mu is equal to 6 .7, the average from last year.

01:05 And because the city manager hopes that we're going to do better and encourages our first responders to do better, our alternative hypothesis is going to be a one -sided, and we want that average to be less than 6 .7.

01:23 So these will be our hypothesis for the significance test to determine whether the average response team time has decreased.

01:34 Now, running a test and comparing your p value with alpha, sometimes we can make errors.

01:42 A type 1 error is when you reject the null hypothesis when it's actually true.

01:49 So in this case, rejecting the null hypothesis would mean that we find convincing evidence that the mean response time has decreased when it really has it.

02:05 So we find convincing evidence at the mean response time has decreased when it actually hasn't, or we'll say when it really hasn't.

02:28 Now, a consequence really is that the city may not investigate other ways to reduce mean response time and more people could die.

02:38 So a consequence is that the city thinks it is responding faster than it is.

02:56 Now that's going to be a problem because the city is going to say, hey, we're doing great, but really they aren't doing great...

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