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A recent study \(^{4}\) considered whether dogs could be trained to detect whether 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 \(\mathrm{c}\) was \(z=17.7 .\) Report the \(\mathrm{P}\) -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

Null Hypothesis

We define the null hypothesis \( H_0 \) as the scenario where the dog's ability to detect a cancer sample is equivalent to random guessing. In this context, since there is 1 cancer sample among 5 samples (4 controls), the probability of randomly guessing correctly is \( p = \frac{1}{5} = 0.2 \). Thus, the null hypothesis is: \( H_0: p = 0.2 \).

02

Alternative Hypothesis (Different from Guessing)

For part (b), we consider the alternative hypothesis \( H_a \) where the probability of a correct detection by the dog differs from random guessing. This can be expressed as \( H_a: p eq 0.2 \). This is a two-tailed hypothesis test.

03

Alternative Hypothesis (Greater than Guessing)

For part (c), we set up a one-tailed alternative hypothesis to test if the dogs' success rate in detecting cancer is greater than the random chance. The alternative hypothesis is: \( H_a: p > 0.2 \).

04

Calculating P-value for the Test Statistic

In part (d), the test statistic for the hypothesis \( H_a: p > 0.2 \) is given as \( z = 17.7 \). Using a standard normal distribution table or a statistical software, we can find the p-value for \( z = 17.7 \). Since this \( z \) value is very large, indicating a high success rate compared to random guessing, the p-value is near 0. Therefore, to three decimal places, the p-value is calculated to be \( p = 0.000 \).

05

Interpretation of the P-value

A p-value this low suggests that the observed results would be extremely unlikely under the null hypothesis. Therefore, we reject the null hypothesis \( H_0: p = 0.2 \) in favor of the alternative hypothesis \( H_a: p > 0.2 \). This indicates that the dogs' ability to detect cancer samples 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 serves as a baseline or a default assumption that there is no effect or no difference. In the context of the dogs detecting cancer by scent, the null hypothesis assumes that the dogs' success rate in detecting cancer is purely by random chance.
Here, if there is a set of five samples, one with cancer and four without, the probability of a dog correctly selecting the cancer sample randomly is 0.2, or 20%. This is because there is one favorable outcome in five possible outcomes. Therefore, the null hypothesis is mathematically specified as \( H_0: p = 0.2 \).
By accepting the null hypothesis, we assume that the dogs do not actually detect cancer, but rather their sitting in front of the cancer sample is equivalent to a random guess.

Alternative Hypothesis

The alternative hypothesis presents a scenario that contrasts with the null hypothesis. It posits that any observed effects or differences are not due to randomness. There are two types of alternative hypotheses in our context:

• **Two-tailed hypothesis**: This tests whether the probability of the dogs detecting cancer is different from random guessing (not equal to 0.2). The formulation is \( H_a: p eq 0.2 \). This can lead to results on two sides; just identifying any difference without directionality.


• **One-tailed hypothesis**: This more specifically tests if dogs have a greater probability of success than random guessing. We express this alternative hypothesis as \( H_a: p > 0.2 \). It focuses on proving if the trained ability of dogs is indeed higher than random chance.

Selecting the correct type of hypothesis depends on the specific research question or interest, guiding how the results are interpreted.

P-value

A p-value quantifies the probability of obtaining a test statistic as extreme as the observed one, assuming the null hypothesis is true. It helps in determining the strength of evidence against the null hypothesis.
For the exercise, with a test statistic of \( z = 17.7 \) for the alternative hypothesis \( H_a: p > 0.2 \), the p-value is calculated to be close to 0.000. This extremely low p-value suggests that the probability of observing such a significant result under the null hypothesis is virtually zero.
Therefore, with a small p-value, we reject the null hypothesis in favor of the alternative hypothesis. Such a result implies that the dogs' ability to detect cancer by scent is significantly better than random guessing, bolstering the notion that dogs can be trained to detect cancer more reliably than by random chance.
In hypothesis testing, a common threshold for p-values is 0.05, meaning if the p-value is below 0.05, the results are statistically significant. Here, with a p-value of 0.000, the conclusion is profoundly significant.