Q6. (a) A manufacturer of light bulbs advertises that, on average, its long-life bulb will last for more than 5,000 hours. Assume that the lifetime of a randomly selected bulb of this type has a standard deviation of 400 hours. A statistician took a random sample of 100 bulbs and measured, for each bulb, the amount of time until the bulb burned out. The sample mean is 5,065 hours. 2. Find a 95% confidence interval for the expected number of hours a randomly selected bulb of this type will last. (b) The statistician has drawn another sample of size n. However, the information about the sample mean is lost after obtaining a 1 – a confidence interval for the expected number of hours a randomly selected bulb of this type will last, which is [5, 030, 5, 070]. (i) Find the sample mean. (ii) If the statistician would like to obtain a 90% confidence interval for the expected number of hours a randomly selected bulb of this type will last and keep the maximum deviation from the sample mean to be 50 hours, what should be the minimum value of n? (iii) If the sample of size 1,000 is obtained and the statistician would like to obtain a 99% con- fidence interval for the expected number of hours a randomly selected bulb of this type will last, what is the width of the interval? (iv) If the statistician has obtained the sample with size 300 and eventually agree with the light bulbs manufacturer's claim, what is the maximum confidence coefficient the statistician applied? (c) The statistician found that the assumption on the standard deviation is incorrect. Therefore, the standard deviation is now an unknown. Find a 95% confidence interval for the expected number of hours a randomly selected bulb of this type will last if (i) the sample size, sample mean, the second sample moment are 100, 5,065 hours and 25814225 respectively. (ii) the statistician took a random sample with size 10,000 and found out that the sample mean and the sample standard deviation are 5,065 and 400 hours respectively.
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.
B. ii, iii and C
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![Q6. (a) A manufacturer of light bulbs advertises that, on average, its long-life bulb will last for more than
5,000 hours. Assume that the lifetime of a randomly selected bulb of this type has a standard
deviation of 40o hours. A statistician took a random sample of 100o bulbs and measured, for
each bulb, the amount of time until the bulb burned out. The sample mean is 5,065 hours.
Find a 95% confidence interval for the expected number of hours a randomly selected bulb of
this type will last.
(b) The statistician has drawn another sample of size n. However, the information about the sample
mean is lost after obtaining a 1 – a confidence interval for the expected number of hours a
randomly selected bulb of this type will last, which is [5, 030, 5, 070].
(i) Find the sample mean.
(ii) If the statistician would like to obtain a 90% confidence interval for the expected number
of hours a randomly selected bulb of this type will last and keep the maximum deviation
from the sample mean to be 50 hours, what should be the minimum value of n?
(iii) If the sample of size 1,000 is obtained and the statistician would like to obtain a 99% con-
fidence interval for the expected number of hours a randomly selected bulb of this type
will last, what is the width of the interval?
(iv) If the statistician has obtained the sample with size 3o0 and eventually agree with the light
bulbs manufacturer's claim, what is the maximum confidence coefficient the statistician
applied?
(c) The statistician found that the assumption on the standard deviation is incorrect. Therefore,
the standard deviation is now an unknown. Find a 95% confidence interval for the expected
number of hours a randomly selected bulb of this type will last if
(i) the sample size, sample mean, the second sample moment are 100, 5,065 hours and 25814225
respectively.
(ii) the statistician took a random sample with size 10,000 and found out that the sample mean
and the sample standard deviation are 5,065 and 400 hours respectively.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F18f8660f-48dc-4e5a-972e-f0de82ccda7c%2F27e11e99-7736-4a53-9d46-56f424bcc35e%2F9v45p6o_processed.png&w=3840&q=75)
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