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Example \(4,\) on whether dogs can detect bladder cancer by selecting the correct urine specimen (out of seven), used the normal sampling distribution to find the P-value. The normal distribution P-value approximates a P-value using the binomial distribution. That binomial P-value is more appropriate when either expected count is less than \(15 .\) In Example \(4, n\) was \(54,\) and 22 of the 54 selections were correct. a. If \(\mathrm{H}_{0}: p=1 / 7\) is true, \(X=\) number of correct selections has the binomial distribution with \(n=54\) and \(p=1 / 7\). Why? b. For \(\mathrm{H}_{a}: p>1 / 7,\) with \(x=22,\) the small sample \(\mathrm{P}\) -value using the binomial is \(\mathrm{P}(22)+\mathrm{P}(23)+\cdots+\mathrm{P}(54)\), where \(\mathrm{P}(x)\) denotes the binomial probability of outcome \(x\) with \(p=1 / 7\). (This equals \(0.0000019 .\) ) Why would the P-value be this sum rather than just \(\mathrm{P}(22) ?\)

Step by step solution

01

Understand the Null Hypothesis

The null hypothesis  is such that the probability of a correct selection by a dog, denoted as \(p\), is equal to \(\frac{1}{7}\), which is the probability of randomly selecting the correct urine sample from seven options. Thus, if \(\mathrm{H}_0:\, p = \frac{1}{7}\), then \(X\), the number of correct selections, follows a binomial distribution with parameters \(n=54\) and \(p=\frac{1}{7}\). This is because there are 54 independent trials and each has a 1/7 probability of success.

02

Define the Alternative Hypothesis

The alternative hypothesis \(\mathrm{H}_a:\, p > \frac{1}{7}\) suggests that the probability of dogs correctly choosing the urine specimen is greater than random chance (\(\frac{1}{7}\)). Thus, \(x = 22\) correct selections observed in the sample needs to be tested against this hypothesis.

03

Compute Binomial Probability

To find the probability that 22 or more of the 54 selections are correct, we sum the probabilities of all outcomes from \(x=22\) to \(x=54\). Each specific \(\mathrm{P}(x)\) can be computed using the binomial probability formula: \[\mathrm{P}(X = x) = \binom{54}{x} \left(\frac{1}{7}\right)^x \left(\frac{6}{7}\right)^{54-x}\]where \(\binom{54}{x}\) is the binomial coefficient.

04

Explanation of P-value

The calculated P-value is the probability that the observed result, or any more extreme result, can occur when the null hypothesis is true. Since we observe 22 correct selections, we calculate \(\mathrm{P}(22) + \mathrm{P}(23) + \cdots + \mathrm{P}(54)\) to determine how often we would see 22 or more successes under the null hypothesis. \(\mathrm{P}(22)\) alone would not account for more extreme results, hence the need to consider all higher successes. This prevents us from concluding \(\mathrm{H}_a\) prematurely if higher results are also notable.

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Key Concepts

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

Null Hypothesis

The concept of the null hypothesis is foundational in statistics. It represents a statement that there is no effect or difference in a situation being tested. In the context of this exercise, the null hypothesis \( H_0: p = \frac{1}{7} \), asserts that the probability \(p\) of dogs making a correct selection purely by chance, is equal to \(\frac{1}{7}\). This is because, when choosing randomly from seven options, each option has an equal chance of being chosen.

• The null hypothesis assumes that any observed effect is due to random variation.
• In hypothesis testing, we often seek to reject the null hypothesis, showing that an actual effect exists beyond random chance.
• To do so, we compare the observed data against what we would expect to see if the null hypothesis were true.

Understanding the null hypothesis helps establish a baseline expectation when analyzing experimental results.

Alternative Hypothesis

Complementing the null hypothesis is the alternative hypothesis, noted as \( H_a: p > \frac{1}{7} \), which suggests that there is a noticeable effect or difference present. Specifically, in this exercise, the alternative hypothesis proposes that dogs can indeed detect bladder cancer better than random chance, which implies a higher probability of correct selections than \(\frac{1}{7}\).

• The alternative hypothesis is what researchers aim to prove—indicative of a real effect or a difference.
• When we find sufficient evidence against the null hypothesis, we may conclude that the alternative hypothesis is more likely to be valid.
• In practice, if statistical tests show the data significantly deviate from what would be expected under the null hypothesis, this strengthens the case for the alternative hypothesis.

The difference between null and alternative hypotheses underpins the essence of hypothesis testing in statistical analysis.

P-value Calculation

The P-value is a critical concept in hypothesis testing as it helps us determine the strength of evidence against the null hypothesis. For the given scenario, the P-value is the probability of observing 22 or more correct selections if the null hypothesis is true. Instead of just calculating \( P(22) \), we sum up all probabilities from \( P(22) \)to \( P(54) \).This cumulative sum reflects the chance of getting 22 or more successes purely by chance under the null hypothesis.

• The P-value helps quantify the unpredictability of our observation under the null hypothesis.
• A small P-value (typically less than 0.05) indicates that such an outcome is rare under the null hypothesis, leading us to question its validity.
• However, a P-value does not measure the magnitude of the effect or its practical significance, only its statistical significance.

In this example, calculating the P-value using a binomial distribution is apt due to the discrete nature of the data and gives a precise measure for hypothesis testing.