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A sample of dogs were trained using a "Do as I do" method, in which the dog observes the trainer performing a simple task (such as climbing onto a chair or touching a chair) and is expected to perform the same task on the command "Do it!" In a separate training session, the same dogs were trained to lie down regardless of the trainer's actions. Later, the trainer demonstrated a new simple action and said "Do it!" The dog then either repeated the new action, or repeated a previous trained action (such as lying down). The dogs were retested on the new simple action after one minute had passed, and after one hour had passed. A "success" was recorded if a dog performed the new simple action on the command "Do it!" before performing a previously trained action. The article "Your dog remembers more than you think" (Science, November \(23,2016,\) www.sciencemag.org/news \(/ 2016 / 11 /\) your -dog-remembers-more-you-think, retrieved May 6,2017 ) reports that dogs trained using this method recalled the correct new action in 33 out of 35 trials. Suppose you want to use the data from this study to determine if more than half of all dogs trained using this method would recall the correct new action. a. Explain why the data in this example should not be analyzed using a large- sample hypothesis test for one population proportion. b. Perform an exact binomial test of the null hypothesis that the proportion of all dogs trained using this method who would perform the correct new action is \(0.5,\) versus the alternative hypothesis that the proportion is greater than \(0.5 .\)

Step by step solution

01

Explain why a large-sample hypothesis test for one population proportion is not appropriate

A large-sample hypothesis test for one population proportion relies on the Central Limit Theorem, where the sample size is large enough for the sampling distribution of the sample proportion to be normally distributed. The rule of thumb is that both the expected success (np) and expected failure (nq) counts be greater than or equal to 10, where n is the number of trials, and p and q are the null hypothesis proportion and its complement, respectively. However, in this case, we have only 35 trials, and the null hypothesis proportion is 0.5. So the expected success and failure counts are 17.5 and 17.5, respectively, which are not very large and close to the threshold of 10. Consequently, relying on a large-sample hypothesis test for one population proportion might not provide accurate results here.

02

State the null and alternative hypotheses

We want to test if more than half of all dogs trained using this method would recall the correct new action. So, the null and alternative hypotheses are: Null hypothesis (H0): The proportion of dogs successfully recalling the correct new action is 0.5 (p=0.5) Alternative hypothesis (H1): The proportion of dogs successfully recalling the correct new action is greater than 0.5 (p>0.5)

03

Perform an exact binomial test

An exact binomial test is appropriate for testing the null hypothesis for small sample sizes. The binomial test calculates the probability of observing the given number of successes (or more extreme) under the assumption that the null hypothesis is true. The test statistic is the number of successes: k=33. First, we calculate the binomial probability of observing 33 or more successes out of 35 with a null hypothesis proportion of 0.5: P(X>=33) = P(X=33) + P(X=34) + P(X=35) Using the binomial probability formula: \(P(X=k)=\binom{n}{k}p^{k}(1-p)^{n-k}\) we can calculate the probabilities for k=33, 34, and 35: \(P(X=33)=\binom{35}{33}(0.5)^{33}(0.5)^{2}\) \(P(X=34)=\binom{35}{34}(0.5)^{34}(0.5)^{1}\) \(P(X=35)=\binom{35}{35}(0.5)^{35}(0.5)^{0}\) Calculate each probability and sum them up to find the p-value: \(p = P(X=33)+P(X=34)+P(X=35)\)

04

Assess statistical significance

Compare the p-value calculated in step 3 with a pre-specified significance level (\(\alpha\), usually set at 0.05 or 0.01) to determine if the null hypothesis can be rejected. If the p-value is smaller than \(\alpha\), we can reject the null hypothesis and conclude that the proportion of dogs successfully recalling the correct new action is significantly greater than 0.5. If the p-value is greater than or equal to the specified \(\alpha\), we cannot reject the null hypothesis and conclude that there is not enough evidence to suggest that the proportion of dogs successfully recalling the correct new action is greater than 0.5.

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Key Concepts

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

Hypothesis Testing

Hypothesis testing is a fundamental statistical method used to make decisions based on data. It involves making an initial assumption called the null hypothesis (H0) and determining whether there's enough evidence to reject this assumption in favor of an alternative hypothesis (H1).

In the context of the dog training study, the null hypothesis states that the probability (p) of dogs recalling the correct new action is exactly 0.5. This assumes a 50% chance that a dog correctly performs the new task. The alternative hypothesis suggests that more than half of the dogs (p > 0.5) can remember the new action, implying the method is effective beyond just random chance.

To decide whether to reject the null hypothesis, we conduct a statistical test and calculate a p-value. This p-value represents the probability of observing results as extreme as the observed outcomes if the null hypothesis were true. If the p-value is below a predetermined significance level (typically 0.05), we reject the null hypothesis, suggesting that the observed result is statistically significant. This process helps us make informed conclusions based on sample data.

Population Proportion

Population proportion refers to the fraction of individuals in a population that exhibits a certain characteristic. In statistical terms, it is often denoted by the symbol p.

In the study exercise, the population proportion of interest is the proportion of dogs that successfully recall and perform the correct new action in response to the command "Do it!". We are specifically interested in knowing if this proportion is greater than 0.5, implying training effectiveness beyond random guessing.

To analyze this, we use the number of successful demonstrations (33 out of 35 trials) as our sample proportion. We then assess whether this sample proportion significantly deviates from 0.5, which would indicate that the population proportion is different (greater in this case) from the null hypothesis. This can be achieved using statistical techniques such as binomial testing, which is particularly useful when dealing with dichotomous outcomes (successes vs. failures) like in this case.

Understanding and correctly estimating the population proportion is crucial as it provides insights into the underlying behavior or characteristics of the broader group being studied.

Central Limit Theorem

The Central Limit Theorem (CLT) is an essential concept in statistics that helps in understanding how sample data can approximate the broader population. It states that, given a sufficiently large sample size, the distribution of the sample mean will be approximately normal, regardless of the original distribution of the population.

In hypothesis testing, especially tests involving population proportions, the CLT aids in justifying the use of normal distribution models. However, the CLT is only valid when the sample size is large enough; typically, this means that both expected successes (np) and failures (nq) need to be at least 10.

In the dog training study, the sample size was only 35, which calls into question the validity of directly applying the CLT. With fewer than the recommended number of expected successes and failures under the null hypothesis, the sampling distribution might not be normal. This is why a large-sample test is not suitable, and instead, an exact binomial test is used. This method does not rely on the normality assumption, providing a more accurate analysis for small sample situations, ensuring reliable results irrespective of the sample size limitation.