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Chapter 9: Problem 22

(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.

Short Answer

Expert verified
Null: \( H_0: p = p_0 \). Alt: \( H_a: p > p_0 \). Type II error is more serious, as it may overlook true improvements.

Step by step solution

01

Define the Parameter of Interest

Let \( p \) be the true proportion of times first responders arrive within 8 minutes of a call. This is the parameter of interest that we want to test.
02

State the Null and Alternative Hypotheses

The null hypothesis \( H_0 \) is that the proportion of successful arrivals within 8 minutes is equal to a given value, \( p_0 \). This can be written as \( H_0: p = p_0 \). The alternative hypothesis \( H_a \) is that the proportion is greater than \( p_0 \). Thus, \( H_a: p > p_0 \).
03

Describe a Type I Error

A Type I error occurs if we reject the null hypothesis \( H_0 \) when it is actually true. This means we will conclude that first responders are arriving within 8 minutes more often than before, when in fact, they are not.
04

Describe a Type II Error

A Type II error occurs if we fail to reject the null hypothesis \( H_0 \) when the alternative hypothesis \( H_a \) is true. This means we will conclude that first responders are not improving their arrival times, when in fact, they are arriving more often within 8 minutes.
05

Analyze the Consequences of Type I and II Errors

The consequence of a Type I error is creating a false sense of improvement and possibly neglecting further actions to improve response times. The consequence of a Type II error is missing an opportunity to recognize and celebrate an actual improvement in response efforts.
06

Identify the More Serious Error

In this setting, the more serious error could be situationally dependent. However, typically, a Type II error might be more serious because it could delay necessary celebrations and acknowledgments of improved efforts which can boost morale and motivate further enhancements.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Significance Test
A significance test is a statistical tool used to determine the likelihood that a given hypothesis about a dataset is true. It helps evaluate whether the observed patterns in data are due to chance or some underlying factor. In this context, we use significance tests to investigate if first responders arrive within 8 minutes more frequently than a specified proportion.
  • The goal is to decide if any observed improvement in response times is statistically significant.
  • This involves comparing real-world data against the null hypothesis using a p-value, which shows the probability of observing data as extreme as ours, assuming the null hypothesis is true.
Significance tests help us make informed conclusions about whether our observations are likely due to random chance or a true change in underlying patterns.
Type I Error
A Type I error occurs when we incorrectly reject a true null hypothesis. It's like a false alarm鈥攄etecting a change or effect that doesn't actually exist.
  • In this scenario, it's concluding that first responders are improving their arrival times within 8 minutes when, in reality, there is no improvement.
  • The consequence is that stakeholders might believe progress is made, potentially halting further initiatives to genuinely improve response times.
To minimize the chances of a Type I error, researchers set a significance level (commonly 0.05), which indicates a 5% risk of making this mistake.
Type II Error
A Type II error happens when we fail to reject a false null hypothesis. This means we miss an actual effect or change.
  • In this setting, it refers to incorrectly believing that first responders are not improving their response times, even though they actually are.
  • This error can lead to missed opportunities for recognizing and encouraging improvement.
Reducing the likelihood of a Type II error involves ensuring a study has enough power, which can be increased with larger sample sizes or stronger effects.
Null Hypothesis
The null hypothesis (\( H_0 \)) is a statement assuming no effect or change. It serves as the default or starting assumption in significance testing.
In our exercise:
  • The null hypothesis posits that the proportion of times first responders arrive within 8 minutes is equal to a specific value (\( p_0 \)).
  • This assumption is tested against the actual data collected, with the goal of either rejecting or failing to reject this hypothesis.
The null hypothesis acts as a benchmark, and it can only be rejected with strong statistical evidence.
Alternative Hypothesis
The alternative hypothesis (\( H_a \)) is the statement indicating there is an effect or change. It represents the hypothesis researchers strive to prove during a significance test.
In this case:
  • The alternative hypothesis suggests that first responders arrive within 8 minutes more frequently than the rate specified by (\( p_0 \)).
  • This hypothesis is considered to be true if the evidence against the null hypothesis is strong enough.
A significance test provides the framework by which we can assess whether there is sufficient evidence to support the alternative hypothesis.

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Most popular questions from this chapter

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