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Dogs and cancer A recent study \(^{6}\) considered whether dogs could be trained to detect if a person has lung cancer or breast cancer by smelling the subject's breath. The researchers trained five ordinary household dogs to distinguish, by scent alone, exhaled breath samples of 55 lung and 31 breast cancer patients from those of 83 healthy controls. A dog gave a correct indication of a cancer sample by sitting in front of that sample when it was randomly placed among four control samples. Once trained, the dogs' ability to distinguish cancer patients from controls was tested using breath samples from subjects not previously encountered by the dogs. (The researchers blinded both dog handlers and experimental observers to the identity of breath samples.) Let \(p\) denote the probability a dog correctly detects a cancer sample placed among five samples, when the other four are controls. a. Set up the null hypothesis that the dog's predictions correspond to random guessing. b. Set up the alternative hypothesis to test whether the probability of a correct selection differs from random guessing. c. Set up the alternative hypothesis to test whether the probability of a correct selection is greater than with random guessing. d. In one test with 83 Stage I lung cancer samples, the dogs correctly identified the cancer sample 81 times. The test statistic for the alternative hypothesis in part \(c\) was \(z=17.7\). Report the P-value to three decimal places, and interpret. (The success of dogs in this study made researchers wonder whether dogs can detect cancer at an earlier stage than conventional methods such as MRI scans.)

Step by step solution

01

Identify the probability of random guessing

In this context, the dog is presented with one cancer sample among five total samples (one cancer and four controls). Each selection is random, so the probability, \( p \), of correctly selecting the cancer sample by chance is \( \frac{1}{5} = 0.2 \). This will be the baseline for the null hypothesis.

02

Set up the null hypothesis

The null hypothesis, \( H_0 \), is that the dog's ability to detect cancer is due to random guessing. Therefore, the probability of a dog correctly identifying a cancer sample is \( p = 0.2 \).

03

Set up the two-sided alternative hypothesis

The two-sided alternative hypothesis, \( H_a \), addresses whether the probability of correctly identifying a cancer sample differs from random guessing. This is expressed as \( p eq 0.2 \).

04

Set up the one-sided alternative hypothesis

The one-sided alternative hypothesis, of interest in this study, tests whether the probability of correctly identifying a cancer sample is greater than by chance. This is expressed as \( p > 0.2 \).

05

Determine the P-value for the test statistic

Given a test statistic \( z = 17.7 \) for the one-sided alternative hypothesis \( p > 0.2 \), we look up the corresponding P-value in the standard normal distribution table. A \( z \)-score of 17.7 is extremely high, and the P-value associated with it is essentially 0, far less than any standard significance level (e.g., 0.05).

06

Interpret the P-value

A P-value of essentially 0 indicates that the probability of obtaining a test statistic as extreme as or more extreme than 17.7, if the null hypothesis were true, is virtually nonexistent. This strongly suggests that the dogs' ability to detect cancer is significantly better than random guessing.

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Key Concepts

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

Null Hypothesis

In hypothesis testing, the null hypothesis acts as a starting point for any statistical investigation. It represents a statement of no effect or no difference. In the context of our exercise on dogs detecting cancer, the null hypothesis is articulated as the dogs relying on random guessing to identify cancer. This is based on the probability calculation for random guessing, where the chance of a dog correctly identifying a cancer sample among five samples (one cancer, four controls) is set at 0.2 or 20%.

The null hypothesis, denoted as \( H_0 \), thus assumes that the probability \( p \) of correctly detecting a cancer sample by the dogs is equal to 0.2: \( H_0: p = 0.2 \). This means the dogs do not possess any special skill beyond chance in detecting cancer. Recognizing this concept is crucial for evaluating whether observed results could have happened under the assumption of this hypothesis.

Alternative Hypothesis

Contrary to the null hypothesis, the alternative hypothesis represents what you aim to prove or substantiate through your test. It's the evidence suggesting that the assumed scenario (null hypothesis) is not accurate. In our cancer-detecting dogs' case, we have two types of alternative hypotheses.

• Two-sided Alternative: This hypothesis contests whether the actual probability of identifying a cancer sample differs from random guessing. It's expressed as \( H_a: p eq 0.2 \). It suggests that the dogs may perform differently, whether better or worse, than pure chance.
• One-sided Alternative: More focused, this hypothesis asserts that the dogs' ability is superior to random guessing, implying \( H_a: p > 0.2 \). This hypothesis tests if the dogs can detect cancer more accurately than if they were just guessing.

Choosing between one or two-sided alternatives depends on the experiment's goal. Our study aims at validating if the dogs' detection skills exceed random chance.

P-value

The P-value is a central concept in hypothesis testing that helps determine the strength of the results against the null hypothesis. It provides the probability of obtaining a test statistic as extreme as the one observed, assuming the null hypothesis is correct.

In the dog study, an incredibly high \( z \)-score of 17.7 was observed when testing the one-sided alternative hypothesis \( p > 0.2 \). The corresponding P-value for such a \( z \)-score is essentially 0. A P-value this low indicates that if the null hypothesis were true (dogs guess randomly), the probability of seeing such extreme results would be virtually nonexistent.

Thus, a very low P-value strongly refutes the null hypothesis, suggesting the dogs' cancer detection ability is not due to random guessing, but a true skill.

Statistical Significance

Statistical significance is a term used to denote that the observed results are not likely to have occurred by random chance under the null hypothesis. It often involves comparing the P-value to a predetermined significance level \( \alpha \), commonly set at 0.05.

In the scenario involving cancer-detecting dogs, the P-value was found to be close to 0. This is much lower than any typical significance level one might select, such as 0.05. This suggests that the results are statistically significant.

• Statistical significance implies that the observed effect (dogs detecting cancer) reflects a real difference and is not due to random chance.
• In scientific studies, establishing statistical significance helps confirm that the findings are reliable and can be attributed to actual abilities or effects rather than coincidental random occurrences.

With a statistically significant result, researchers can confidently assert that the dogs are genuinely better than chance at detecting cancer.