Lab 11 Instructions: For this lab, your solutions should be written or typed up neatly and uploaded to Canvas as PDF, document, or image scan. In probability theory, the normal distribution (also known as a Gaussian distribution or bell curve) is a commonly used probability distribution. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known specifically. Describing the normal distribution using a mathematical function (called a probability distribution function) is given: f(x) = 1 O√2π -(x-μ)² e 20² P(a ≤ x ≤ B) = The area underneath a probability density function corresponds with the probability of the event occurring: •f²f(x) dx The main mathematical issue with the normal distribution is that the function fe-x² dx has no elementary solution. So, how can we solve the area under the curve if we can't solve the integral? We must take another approach. In many cases where f has no elementary antiderivative, we are able to take f and convert it into an infinite power series. This is the fundamental idea behind Taylor and Maclaurin series. We will be discussing this approach in lab and class this week. Here is the problem that you will be solving: The lengths of a certain population of Atlantic croaker fish are normally distributed with an average length of μ10 inches and a standard deviation of a = 2 inches. In each part, your goal is to determine how many terms are needed to insure a result accurate to at least 0.1%. 1. Find the probability that a randomly selected croaker fish from this population will be between a = 11 inches and ß = 13 inches long. 2. Find the probability that a randomly selected croaker fish from this population will be less than B 8 inches long (use a = 0). 3. In a statistics course, this type of question can be solved by using a "Z-score" and "Z-table" or by using a "Probability Calculator." Use a probability calculator to check your solutions. (Here is one: https://www.mathportal.org/calculators/statistics-calculator/normal-distribution-calculator.php) +.25 .20 15 +.10 Normal Distribution Function The mean of the distribution The standard deviation 7.05 +-.05 -.10 7 10 11 12 13 ✓ You are encouraged to use wxMaxima to integrate f's Maclaurin series and calculate the probabilities to the desired level of accuracy!