The mean gestational length of a sample of 208 horses is 343.7 days, with a standard deviation of 10.4 days. The data set has a bell-shaped distribution.


b. Determine whether a gestational length of 318.4 days is unusual.

Verified step by step guidance

1. Step 1: Understand the concept of 'unusual' values in a bell-shaped distribution. In statistics, a value is considered unusual if it lies more than 2 standard deviations away from the mean.
2. Step 2: Calculate the lower and upper bounds for usual values using the formula: Lower Bound = Mean - 2 × Standard Deviation, and Upper Bound = Mean + 2 × Standard Deviation. Use the given mean (343.7 days) and standard deviation (10.4 days) in the formula.
3. Step 3: Substitute the values into the formula for the lower bound: Lower Bound = 343.7 - 2 × 10.4. Perform the subtraction and multiplication to find the lower bound.
4. Step 4: Substitute the values into the formula for the upper bound: Upper Bound = 343.7 + 2 × 10.4. Perform the addition and multiplication to find the upper bound.
5. Step 5: Compare the given gestational length of 318.4 days to the calculated lower and upper bounds. If 318.4 days lies outside these bounds, it is considered unusual; otherwise, it is not.

Key Concepts

Mean

The mean is the average value of a data set, calculated by summing all the values and dividing by the number of observations. In this context, the mean gestational length of 343.7 days represents the central tendency of the sample of horses, providing a benchmark against which individual gestational lengths can be compared.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates a wider spread. Here, the standard deviation of 10.4 days helps assess how typical or atypical a gestational length of 318.4 days is in relation to the mean.

Z-Score

A Z-score quantifies how many standard deviations a data point is from the mean. It is calculated by subtracting the mean from the value and then dividing by the standard deviation. In this case, calculating the Z-score for a gestational length of 318.4 days will help determine if it is considered unusual, typically defined as a Z-score less than -2 or greater than 2.