Answer :
Final answer:
We can use the normal distribution to calculate probabilities for body temperatures of healthy dogs, including the chance of temperatures being above or below certain values, and expected numbers of dogs falling within certain temperature ranges.
Explanation:
To solve these probability questions, we can use the normal distribution. Let's calculate the probability in each case:
a) We need to find P(X > 102.12), where X is the body temperature of a healthy dog. We can use the Z-score formula: Z = (X - mean) / standard deviation. Plugging in the values, we get Z = (102.12 - 101.5) / 0.3 = 2.0667. Using a standard normal distribution table, we can find that the probability of Z > 2.0667 is approximately 0.019, or 1.9%.
b) We need to find P(101 < X < 102). Again, we use the Z-score formula to convert the lower and upper limits: Z1 = (101 - 101.5) / 0.3 = -1.6667 and Z2 = (102 - 101.5) / 0.3 = 1.6667. Using the standard normal distribution table, we can find that the probability of -1.6667 < Z < 1.6667 is approximately 0.950, or 95%.
c) Out of 500 dogs, we want to find the expected number of dogs with temperatures less than 100.8. We can use the Z-score formula as before: Z = (100.8 - 101.5) / 0.3 = -2.3333. Using the standard normal distribution table, we can find that the probability of Z < -2.3333 is approximately 0.009, or 0.9%. So, the expected number of dogs would be 500 * 0.009 = 4.5 dogs.
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