Answer :
Final answer:
The chance of a healthy dog's body temperature exceeding 102.12 degrees Fahrenheit is about 2%. The probability that a healthy dog will have a body temperature within the range of 101 to 102 degrees Fahrenheit is approximately 90.54%. Out of 500 dogs, we could expect around 5 dogs to have a body temperature below 100.8 degrees Fahrenheit.
Explanation:
To calculate the chance or probability of a certain body temperature in a dog, we use the properties of the normal distribution. The first step is to standardize the value by converting it into a z-score using the formula (X - μ) / σ, where X is the value, μ is the mean, and σ is the standard deviation. In this scenario, the mean (μ) is 101.5 degrees Fahrenheit and the standard deviation (σ) is 0.3 degrees Fahrenheit.
For the first part, we want the probability that body temperature is more than 102.12 degrees Fahrenheit. The z-score for 102.12 degrees using the formula above would be (102.12 - 101.5) / 0.3. This yields a value of 2.07. Using a standard normal distribution table or a calculator, we find that the area to the left of z = 2.07 is approximately 0.98. Since we are interested in the area to the right (greater than), we subtract this from 1 to get the desired probability. As such, the chance that a healthy dog will have a body temperature of more than 102.12 degrees Fahrenheit is approximately 0.02 or 2%.
For the second part, the temperatures 101 and 102 degrees Fahrenheit correspond to z-scores of -1.67 and 1.67, respectively. The probability that a z-score is between these two values is approximately 0.9054 or 90.54%.
Finally, for the third part, we want to know the expected number of dogs with body temperatures less than 100.8 degrees Fahrenheit out of 500 dogs. This is equivalent to the probability that a dog's body temperature would be less than 100.8 degrees Fahrenheit, multiplied by the total number of dogs (500). The z-score for 100.8 degrees is -2.33. The probability corresponding to this z-score is approximately 0.01 or 1%. Therefore, we would expect approximately 1% of 500, or 5 dogs, to have body temperatures of less than 100.8 degrees Fahrenheit.
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