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 first responders are arriving within 8 minutes of the call more often. 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. (d) If you sustain a life-threatening injury due to a vehicle accident, you want to receive medical treatment as quickly as possible. Which of the two significance tests $-H_{0} : \mu=6.7$ versus $H_{a} : \mu<6.7$ or the one from part (a) of this exercise - would you be more interested in? 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 first responders are arriving within
8 minutes of the call more often. 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.
(d) If you sustain a life-threatening injury due to a vehicle accident, you want to receive medical
treatment as quickly as possible. Which of the two significance tests $-H_{0} : \mu=6.7$ versus $H_{a} : \mu<6.7$ or the one from part (a) of this exercise - would you be more interested in? Justify your answer.

Key Concepts

One-Tailed vs. Two-Tailed Tests

A one-tailed test is used when the research question is directional, such as evaluating whether the response times have improved (i.e., more arrivals within the target time) compared to a set benchmark. In contrast, a two-tailed test is appropriate when deviations in either direction are of interest. In this setting, because the primary concern is whether response times are improved (fewer minutes than before), a one-tailed test is more appropriate.

Type I and Type II Errors

A Type I error occurs when the null hypothesis is rejected when it is actually true, meaning that we might mistakenly conclude that response times have improved when they have not. Conversely, a Type II error happens when we fail to reject the null hypothesis despite the alternative hypothesis being true, potentially overlooking a real improvement in response times. Each error type has direct consequences for decisions in emergency response settings, impacting policy and resource allocation.

Significance Testing

Significance testing is a procedural framework used to decide whether the differences observed in sample data are unlikely to have occurred by random chance under the assumption that the null hypothesis is correct. It involves selecting a significance level, calculating a test statistic from the sample, and comparing it against a critical value to determine if there is sufficient evidence to support the alternative hypothesis.

Parameter of Interest

In hypothesis testing, the parameter of interest is the true proportion of life-threatening accident calls where emergency responders arrive within 8 minutes. This parameter helps quantify the performance of the emergency response system and forms the basis for comparing the observed performance against a historical or target standard.

Null and Alternative Hypotheses

The null hypothesis is a statement of no change or no improvement (for example, that the proportion of timely responses remains at 78%), whereas the alternative hypothesis claims that there has been an improvement (that the proportion of timely responses is greater than 78%). These hypotheses form the foundation of a significance test, providing a framework to evaluate if the evidence from the data supports a true improvement.

Transcript

00:02 In the slow response time problem, we're told that the emergency personnel arrived within eight minutes after 78 % of all calls involving life -threatening injuries last year.

00:16 Again, our city manager shares this information and encourages first responders to do better.

00:22 So in this significance test, we're going to say that our parameter is p, which is the proportion of all calls in which first responders arrived within.

00:32 8 minutes.

00:34 And so in this case, the null hypothesis will be that p is equal to that percentage 0 .78.

00:45 And the alternative hypothesis is that p is going to be greater than 0 .78 because we're investigating when these first responders are arriving.

01:01 So we have our hypothesis for, for the significance test.

01:07 Now let's talk about potential errors.

01:09 We know that a type 1 error is we reject the null hypothesis when it's true.

01:16 So in this case, by rejecting the null hypothesis, we would find that convincing evidence, convincing evidence that the proportion of all calls in which first responders arrived within eight minutes had increased when it really didn't.

01:34 So again, the type 1 error, would be that we find convincing evidence at the proportion of all calls in which first responders arrived within eight minutes had decreased, oh, had increased, i'm sorry, when it really hasn't.

02:09 Now, a consequence might be is that the city may not investigate other ways to reduce the response time and more people could die.

02:20 So the city might not try to better the response time and people could die, ultimately.

02:34 So again, this is saying we reject the null hypothesis when it's true.

02:38 We find convincing evidence that the proportion of calls in which first responders arrived within eight minutes had increased.

02:46 So like we think that they're getting better, when the consequences of this type 1 error might be that the city isn't going to try to better the response time, and people could be dying...

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