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Chapter 6: Problem 25

According to dogtime .com, the mean weight of an adult St. Bernard dog is 150 pounds. Assume the distribution of weights is Normal with a standard deviation of 10 pounds. a. Find the standard score associated with a weight of 170 pounds. b. Using the Empirical Rule and your answer to part a, what is the probability that a randomly selected St. Bernard weighs more than 170 pounds? c. Use technology to confirm your answer to part \(\mathrm{b}\) is correct. d. Almost all adult St. Bernard's will have weights between what two values?

Short Answer

Expert verified
a. The standard score associated with a weight of 170 pounds is 2.0. b. The probability that a randomly selected St. Bernard weighs more than 170 pounds is 2.5%. c. Using technology (like a Z-score calculator), we can confirm our calculated probability. d. Almost all adult St. Bernard's will have weights between 120 and 180 pounds.

Step by step solution

01

Calculating the Standard Score or Z-score

Subtract the mean weight from the given weight (170 - 150) and divide the result by the standard deviation: \(Z = (170 - 150) / 10 = 2.0\)
02

Using the Empirical Rule to Determine Probability

Given that our calculated Z-score is 2.0, and according to the Empirical Rule, approximately 95% of the weights are within 2 standard deviations of the mean, then only 5% would be more than 2 standard deviations away. Since we are interested in weights greater than 170 pounds, and considering the symmetry of a normal distribution, the proportion greater than 2 standard deviations is 5%/2 = 2.5%.
03

Validation Using Technology

At this stage, it would be instructive to use an online Z-score calculator to confirm our calculations. Pour the values into a Z-score calculator: mean = 150, standard deviation = 10, and the X value = 170. The calculator should return a probability of 0.025 or 2.5%, confirming our calculations.
04

Define the Weight Range for Almost All Adult St. Bernard's

Since almost 99.7% of the weights fall within 3 standard deviations of the mean (according to the empirical rule), we need to calculate the weights that are 3 standard deviations away from the mean. For weights less than the mean, calculate 150 - 3*10 = 120 lbs. For weights more than the mean, calculate 150 + 3*10 = 180 lbs. So, almost all adult St. Bernard's will have weights between 120 and 180 pounds.

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