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Chapter 6: Problem 26

Whales have one of the longest gestation periods of any mammal. According to whalefacts.org, the mean gestation period for a whale is 14 months. Assume the distribution of gestation periods is Normal with a standard deviation of \(1.2\) months. a. Find the standard score associated with a gestational period of \(12.8\) months. b. Using the Empirical Rule and your answer to part a, what percentage of whale pregnancies will have a gestation period between \(12.8\) months and 14 months? c. Would it be unusual for a whale to have a gestation period of 18 months? Why or why not?

Short Answer

Expert verified
a. The Z-score for a 12.8 month gestation period is approximately -1. b. About 34% of pregnancies have a gestation period between 12.8 and 14 months. c. Using a Z-score calculation, we can assess whether an 18 month gestation period is considered 'unusual' based on how many standard deviations it is from the mean.

Step by step solution

01

Calculate the Z-score for 12.8 months gestation period

To calculate the Z-score, which indicates how many standard deviations an element is from the mean, use the formula :\[ Z = \frac {X - \mu} {\sigma} \]Where \(X\) is the value we are interested in (12.8 months in this case), \(\mu\) is the mean of the distribution (14 months) and \(\sigma\) is the standard deviation (1.2 months). Substituting values into the formula, we have:\[Z = \frac {12.8 - 14} {1.2} \]
02

Use the empirical rule and Z-score for calculating percentage

The Empirical Rule states that for a normal distribution, almost all data will fall within three standard deviations of the mean. Here, we need to calculate the percentage of pregnancies that will have a gestation period between 12.8 months and 14 months.Based on our calculation from Step 1, we find that 12.8 months is less than a standard deviation away from the mean. And as per the Empirical Rule, approximately 34% of the distribution lies between the mean and one standard deviation below the mean. Therefore, about 34% of the pregnancies have a gestation period between 12.8 months and 14 months.
03

Assess if a gestation period of 18 months is unusual

For a value to be considered 'unusual', it typically needs to be more than 2 or 3 standard deviations away from the mean. To check if the 18 months period is unusual, calculate the corresponding Z-score:\[Z = \frac {18 - 14} {1.2} \]Compare the resulting Z-score to 2 or 3. If it is greater, we can consider the value as 'unusual'. If it is less, it isn't considered 'unusual'.

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