(a) A 100(1- a)% confidence interval for the mean u of a normal population when the value of a is known is given by Under the same conditions as those leading to the interval, x = µ) < 1.645= 0.95. Use this to derive a one-sided interval for u that has infinite width and provides a lower confidence bound on u. x - 1.645 • (F - 100) What is this interval for the true average porosity of a certain seam of coal samples that is normally distributed with true standard deviation 0.76 if the average porosity for 20 specimens from the seam was 4.35? (If you need to use co or -0o, enter INFINITY or INFINITY, respectively. Round your answers to four decimal places.)
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.
Please solve both (a) & (c) and show/explain the steps, thanks!!
![### Confidence Interval for the Mean
**a) A 100(1 – α)% confidence interval for the mean μ of a normal population when the value of σ is known is given by**
\[
\left( \bar{x} - z_{\alpha/2} \frac{\sigma}{\sqrt{n}}, \bar{x} + z_{\alpha/2} \frac{\sigma}{\sqrt{n}} \right).
\]
**Under the same conditions as those leading to the interval,**
\[
P\left( \frac{\bar{x} - \mu}{\left(\frac{\sigma}{\sqrt{n}}\right)} < 1.645 \right) = 0.95.
\]
**Use this to derive a one-sided interval for μ that has infinite width and provides a lower confidence bound on μ.**
1. \((0, \bar{x} = 1.645\frac{\sigma}{\sqrt{n}})\)
2. \(\left(\bar{x} - 1.645\frac{\sigma}{\sqrt{n}}, \infty\right)\)
3. \(\left(0, \bar{x} + 1.645\frac{\sigma}{\sqrt{n}}\right)\)
4. \(\left(\bar{x} = 1.645\frac{\sigma}{\sqrt{n}}, \infty\right)\)
5. \(\left(\bar{x} + 1.645\frac{\sigma}{\sqrt{n}}, \infty\right)\)
**Correct Answer:**
- \(\left(\bar{x} - 1.645\frac{\sigma}{\sqrt{n}}, \infty\right)\) (Option 2)
### Problem
What is this interval for the true average porosity of a certain seam of coal samples that is normally distributed with true standard deviation 0.76 if the average porosity for 20 specimens from the seam was 4.35? (If you need to use ∞ or −∞, enter INFINITY or −INFINITY, respectively. Round your answers to four decimal places.)
**Answer Boxes:**
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![### Educational Transcription
#### Problem Statement
(b) Generalize the result of part (a) to obtain a lower bound with confidence level 100(1 − α)%.
Options:
1. \( \left( -\overline{x} + z_\alpha \frac{\sigma}{\sqrt{n}}, \infty \right) \)
2. \( \left( -\infty, \overline{x} - z_\alpha \frac{\sigma}{\sqrt{n}} \right) \)
3. \( \left( -\infty, \overline{x} + z_\alpha \frac{\sigma}{\sqrt{n}} \right) \)
4. \( \left( \overline{x} - z_\alpha \frac{\sigma}{\sqrt{n}}, \infty \right) \) [Correct]
5. \( \left( \overline{x} - z_{\alpha/2}, \overline{x} + z_\alpha \frac{\sigma}{\sqrt{n}} \right) \)
(c) What is an analogous interval to that of part (b) that provides an upper bound on μ?
Options:
1. \( \left( -\overline{x} + z_\alpha \frac{\sigma}{\sqrt{n}}, \infty \right) \)
2. \( \left( -\infty, \overline{x} - z_\alpha \frac{\sigma}{\sqrt{n}} \right) \)
3. \( \left( -\infty, \overline{x} + z_\alpha \frac{\sigma}{\sqrt{n}} \right) \) [Correct]
4. \( \left( \overline{x} - z_\alpha \frac{\sigma}{\sqrt{n}}, \infty \right) \)
5. \( \left( \overline{x} - z_{\alpha/2}, \overline{x} + z_\alpha \frac{\sigma}{\sqrt{n}} \right) \)
#### Analysis
Compute this 99% interval for the true average stray-load loss \( \mu \) (watts) for a certain type of induction motor when the line current is held at 10 amps for a speed of 1,500 rpm, when \( n = 25 \) and \( \overline{x} = 58.3 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F56945fae-aeca-4f60-8c3d-a266c334f238%2Fcd2f7a91-6c2f-4811-b8ff-3ddb92815d95%2Fk9ccpa_processed.png&w=3840&q=75)
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