Proportions A study selected independent random samples of 50 dogs and 60 cats and found that 38 of the dogs and 15 of the cats could be trained to fetch a ball. Do these data provide convincing evidence that the proportion of all dogs that can be trained to fetch a ball is greater than the proportion of all cats that can be trained to fetch a ball? Use \( a=0.05 \). Calculate the test statistic and P-value for this test, and then complete this concluding statement. If needed, use the z-table to answer the question. Because the P-value of \( \square \) is \( \square \) \( a=0.05 \) we \( \square \) \( \mathrm{H}_{0} \) We \( \square \) convincing evidence to conclude that the proportion of all dogs that can be trained to fetch a ball is \( \square \) the proportion of all cats that can be trained to fetch a balf.

To determine whether the proportion of dogs that can fetch is greater than that of cats, we start by calculating the sample proportions: - For dogs: \( \hat{p}_d = \frac{38}{50} = 0.76 \) - For cats: \( \hat{p}_c = \frac{15}{60} = 0.25 \) Next, we calculate the pooled proportion and the standard error: - Pooled proportion: \( \hat{p} = \frac{38 + 15}{50 + 60} = \frac{53}{110} \approx 0.4818 \) - Standard error (SE): \( SE = \sqrt{\hat{p}(1 - \hat{p}) \left( \frac{1}{n_d} + \frac{1}{n_c} \right)} = \sqrt{0.4818(0.5182) \left( \frac{1}{50} + \frac{1}{60} \right)} \approx 0.0912 \) The z-test statistic is then calculated as: - \( z = \frac{\hat{p}_d - \hat{p}_c}{SE} = \frac{0.76 - 0.25}{0.0912} \approx 5.564 \) Using a z-table, we find the P-value corresponding to \( z = 5.564 \) is extremely small (essentially 0). Now we can complete the conclusion: Because the P-value of \( 0 \) is less than \( a = 0.05 \), we reject \( \mathrm{H}_{0} \). We have convincing evidence to conclude that the proportion of all dogs that can be trained to fetch a ball is greater than the proportion of all cats that can be trained to fetch a ball. --- Training animals, especially dogs, has a long history dating back to ancient civilizations. They were often bred for specific traits such as herding, hunting, or companionship, leading to varying levels of trainability. The ability to fetch is a prime example of how dogs’ inherent behaviors can be harnessed through training, showcasing their loyalty and eagerness to please. In real-world applications, analyzing animal behavior through proportional statistics can assist trainers, breeders, and even pet owners in understanding which breeds might yield more success in training. This can positively impact their selection process and overall pet-owner experience, emphasizing the importance of matching the pet's characteristics with the owner's expectations and lifestyle.