**Normal Distribution** 2. You have a normal distribution with a mean \( \mu = 54 \) and a standard deviation \( \sigma = 7 \). A. **Label the axis for this normal distribution.** A normal distribution curve is drawn with x-axis values labeled as 33, 40, 47, 54, 61, 68, and 75. B. **95% of the data falls between 40 and 68.** C. **What is the probability of getting a value between 45 and 62? Show work.** \[ P(45 \leq X \leq 62) = P\left(\frac{45 - 54}{7} \leq Z \leq \frac{62 - 54}{7}\right) \] Calculation gives a probability of 0.7744. D. **What is the probability of getting a value above 58? Show work.** Calculation involves finding \( P(Z \geq 0.57) \) using the standard normal table, leading to a probability value (annotations show sub-steps including Z-value transformation). E. **What observation cuts off the top 10% of the data? Show work.** Calculation involves finding the Z-score that corresponds to the 90th percentile and applying it to the normal distribution equation. 3. The length of time it takes for a manufacturer to build a car is normally distributed with a mean of 20 hours and a standard deviation of 1.5 hours. Use this information to answer the following questions. A. **What percent of cars are made in less than 17 hours? Show work.** Calculations would involve finding \( P(X < 17) \) using the Z-transform and standard normal distribution table.
Family of Curves
A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.
Euclidean Geometry
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.
Lines and Angles
In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.
Need help with D and E
![**Normal Distribution**
2. You have a normal distribution with a mean \( \mu = 54 \) and a standard deviation \( \sigma = 7 \).
A. **Label the axis for this normal distribution.**
A normal distribution curve is drawn with x-axis values labeled as 33, 40, 47, 54, 61, 68, and 75.
B. **95% of the data falls between 40 and 68.**
C. **What is the probability of getting a value between 45 and 62? Show work.**
\[
P(45 \leq X \leq 62) = P\left(\frac{45 - 54}{7} \leq Z \leq \frac{62 - 54}{7}\right)
\]
Calculation gives a probability of 0.7744.
D. **What is the probability of getting a value above 58? Show work.**
Calculation involves finding \( P(Z \geq 0.57) \) using the standard normal table, leading to a probability value (annotations show sub-steps including Z-value transformation).
E. **What observation cuts off the top 10% of the data? Show work.**
Calculation involves finding the Z-score that corresponds to the 90th percentile and applying it to the normal distribution equation.
3. The length of time it takes for a manufacturer to build a car is normally distributed with a mean of 20 hours and a standard deviation of 1.5 hours. Use this information to answer the following questions.
A. **What percent of cars are made in less than 17 hours? Show work.**
Calculations would involve finding \( P(X < 17) \) using the Z-transform and standard normal distribution table.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0ed02b96-00f1-4605-bbb0-113e667487ea%2F982df827-eadb-4b80-a366-b868aec29ec2%2Femjgc.jpeg&w=3840&q=75)
Let X be a random variable, such that, X~ N(54,72)
D.)
Hence, the probability of getting a value above 58 is 0.284.
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