Random Variables and Distributions
Normal distribution: Finding a probability, basic
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Kennedce

Suppose that the antenna lengths of woodlice are approximately normally distributed with a mean of 0.25 inches and a standard deviation of 0.05 inches. What proportion of woodlice have antenna lengths that are more than 0.23 inches? Round your answer to at least four decimal places.

To find the proportion of woodlice with antenna lengths more than 0.23 inches, we first convert the length to a z-score using the formula: . Here, , , and . So, . Using standard normal distribution tables or a calculator, we find . Therefore, the probability of antenna lengths more than 0.23 inches is . Rounded, that gives us approximately 0.6554.

For a bit of historical context, the normal distribution was first introduced by the mathematician Carl Friedrich Gauss in the early 19th century. He used it to describe the distribution of errors in astronomical observations. This “bell curve” beautifully models various real-world phenomena, from heights of individuals to measurement errors, making it a fundamental pillar of statistics!

If you’re venturing deeper into this topic, consider exploring the Central Limit Theorem (CLT). This theorem states that the means of sufficiently large random samples from a population will be approximately normally distributed, regardless of the original population’s distribution. It’s a game-changer in statistics, as it allows researchers to infer properties about populations even when they’re dealing with non-normal data!