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Z-score is a vital statistic that indicates how far a data point is from the mean in terms of standard deviations. It's a standard measure that tells us how unusual or typical a particular observation is compared to the dataset as a whole.
For example, in our given problem, the gestation period dataset for hippos has a mean of 270 days and a standard deviation of 7 days.

To calculate the z-score, you'll use the formula:

• Z-score: \( z = \frac{(x - μ)}{σ} \)
• Where \( x \) is the data point in question, \( μ \) is the mean, and \( σ \) is the standard deviation.

If we want to know how many standard deviations a gestation period of 260 days is from the mean:

• the calculation becomes \( (260 - 270) / 7 = -1.43 \).

A negative z-score implies the data point is below the mean, as seen in this example.
This helps us understand that 260 days is 1.43 standard deviations below the mean gestation period for hippos.

Gestation period statistics can give us interesting insights into the reproductive patterns of species like hippos.
For hippopotami, with a mean gestation period of 270 days, most pregnancies will fall within a certain range around this mean.
The standard deviation, which is 7 days in this case, tells us about the variability of these gestation periods.

By employing statistical methods, we can solve questions such as:

• What percentage of hippos will have a gestation period less than a specific timeframe, say 260 days?
• How long is the gestation period if only 6% of hippos have longer pregnancies?

These inquiries rely on the characteristics of the normal distribution where the mean and standard deviation are used to measure probabilities. For instance, knowing that 6% have periods longer than 258.8 days shows us how statistics can set expectations for the anomalies and variations in biological phenomena.

The standard normal distribution is a special case of the normal distribution with a mean (μ) of zero and a standard deviation (σ) of one.
Standard normal distribution tables or z-tables are used to find probabilities and percentiles related to z-scores.

In our hippo exercise, when we calculated a z-score of -1.43 for a gestation period of 260 days, the z-table helped us understand what proportion of hippos have shorter gestation periods.
A look at the z-table indicates 7.64% of hippos fall below this mark.

Moreover, the standard normal distribution helps identify extremes, such as Fiona the Hippo's early birth.
With her gestation period of 228 days resulting in a z-score of -6, we find that this is an extreme outlier, as most values fall within 3 standard deviations of the mean.

• This implies that occurrences like Fiona's are extremely rare, showcased by the fact that less than 0.1% of hippos have similarly short gestations.

The familiarity with the standard normal distribution is essential for interpreting these types of statistical outcomes.