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A group of veterinarians wants to test a new canine vaccine for Lyme disease. (Lyme disease is transmitted by the bite of an infected deer tick.) In an area that has a high incidence of Lyme disease, 100 dogs are randomly selected (with their owners' permission) to receive the vaccine. Over a 12 -month period, these dogs are periodically examined by veterinarians for symptoms of Lyme disease. At the end of 12 months, 10 of these 100 dogs are diagnosed with the disease. During the same 12 -month period, \(18 \%\) of the unvaccinated dogs in the area have been found to have Lyme disease. Let \(p\) be the proportion of all potential vaccinated dogs who would contract Lyme disease in this area. a. Find a \(95 \%\) confidence interval for \(p\). b. Does \(18 \%\) lie within your confidence interval of part a? Does this suggest the vaccine might or might not be effective to some degree? c. Write a brief critique of this experiment, pointing out anything that may have distorted the results or conclusions.

Step by step solution

01

Calculate the Sample Proportion

This is done by dividing the number of dogs that got Lyme disease by the total number of dogs in the sample. So, the sample proportion (p̂) is \(10/100 = 0.1\) or \(10% \).

02

Calculate the Confidence Interval

For a 95% confidence interval, the value of z is 1.96 (from the z-table). The formula for the confidence interval is: \[p̂ ± z * sqrt((p̂*(1-p̂))/n)\], where n is the size of the sample. Substituting the values in, we get \[0.1 ± 1.96 * sqrt((0.1*0.9)/100)\]. Simplifying this expression gives us the interval (0.04, 0.16).

03

Compare the Unvaccinated Dogs' Percentage

The percentage of unvaccinated dogs getting Lyme disease is 18%. This value does not lie within the confidence interval calculated in step 2 (0.04, 0.16), which means the vaccine might indeed be effective.

04

Critique of the Experiment

Many factors may distort the results of this experiment. For example, the sample size of 100 dogs may not be representative enough of all potential vaccinated dogs, especially since the incidence of the disease is high in the area. Moreover, the period of 12 months might be insufficient, as the infection may show on a longer time scale. Random selection of the dogs is mentioned, which is a good aspect, but information is missing about the age, race, or state of health of the dogs, which could be confounding factors.

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Key Concepts

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

Sample Proportion

The sample proportion is a key concept in statistics used to estimate the proportion of a certain trait or outcome within a population. In this case, the sample proportion tells us how many vaccinated dogs still contracted Lyme disease from the group that was tested. To find this, we divide the number of affected dogs by the total number in the sample. Here, out of 100 vaccinated dogs, 10 were diagnosed with Lyme disease. By dividing 10 by 100, we calculate the sample proportion to be 0.1 or 10%.
This proportion helps us to understand how common Lyme disease is among the vaccinated population in this high-risk area, providing a basis for further statistical analysis.

Z-Score

The z-score is a statistical measure that describes a value's position relative to the mean of a group of values, measured in standard deviations. In studying confidence intervals, the z-score is used to determine how many standard deviations a data point is from the mean.
In our Lyme disease vaccine study, we use a z-score to calculate the 95% confidence interval to predict the true proportion of vaccinated dogs that might contract the disease. For a 95% confidence interval, the z value is typically 1.96. We use the formula \[p̂ ± z \times \sqrt{(p̂(1-p̂)/n)}\]where \(p̂\) is the sample proportion, and \(n\) is the sample size. Plugging in our values, \[0.1 ± 1.96 \times \sqrt{(0.1 \times 0.9)/100} = (0.04, 0.16)\]This interval helps us estimate, with 95% confidence, the proportion of all vaccinated dogs that might get Lyme disease.

Critique of Experiment

Critiquing the experiment involves assessing factors that could affect the reliability and validity of the results. First, the sample size of 100 dogs may not be large enough to generalize the findings to all vaccinated dogs in areas with high disease incidence. A larger sample may provide more accurate results.
Secondly, the duration of the study was 12 months. Lyme disease symptoms might develop after this period, potentially affecting the findings. Additionally, the study did mention random selection of dogs, but it didn't account for other factors such as age, genetic breed predispositions, or immune health, which might impact susceptibility to Lyme disease.

• A more diverse sample could offer clearer insights.
• Longer observation might reflect true vaccine efficacy.
• Including various dog breeds and health conditions could improve the study's outcomes.

Lyme Disease Vaccine Study

This Lyme disease vaccine study is designed to evaluate the effectiveness of a new vaccine in preventing the disease among dogs. Lyme disease, transmitted via infected deer ticks, is a significant concern in areas where it's prevalent. Vaccinating canines is a proactive approach to reducing infection rates.
This study involved 100 dogs randomly selected to receive the vaccine, with a follow-up period of 12 months to monitor for disease symptoms. At the end of the study, 10% of vaccinated dogs contracted Lyme disease, whereas 18% of unvaccinated dogs were diagnosed with it.

• The lower infection rate in vaccinated dogs suggests some level of vaccine effectiveness.
• The study's results are crucial for determining broader vaccine distribution.
• Understanding these outcomes helps vets and pet owners make informed health decisions for dogs in Lyme-prone areas.

By evaluating this study's design and results, we gain better insights into preventing Lyme disease in the canine population.