Question

**Problem 7**

Suppose you have a random variable X that represents the
lifetime of a certain brand of light bulbs. Assume that the
lifetime of light bulbs are approximately normally distributed with
mean 1400 and standard deviation 200 (in other words X ~ N(1400,
200^{2})).

Answer the following using the standard normal distribution table:

Approximate the probability of a light bulb lasting less than 1250 hours.

Approximate the probability that a light bulb lasts between 1360 to 1460 hours.

Approximate the probability that a light bulb lasts more than 1500 hours.

Draw this distribution’s shape. Note the value of X that is in the middle of the distribution and use the empirical rule to approximate the values that contain the middle 68% and 95% of all observations.

Find the five-number summary of light bulb lifetimes (approximate A- and A+ by finding the outlier criteria based on the z-scores of the quantiles).

f) What is the probability that 225 randomly selected lightbulbs will have an average within 25 hours to the actual population mean?

Answer #1

The life of light bulbs is distributed normally. The variance of
the lifetime is 225 and the mean lifetime of a bulb is 520 hours.
Find the probability of a bulb lasting for at most 533 hours. Round
your answer to four decimal places.

Let X denote the lifetime of a light bulb of a certain brand.
Suppose that the lifetime of a light bulb of this particular brand
has an exponential distribution with mean lifetime of 200 hours.
What is the probability that a light bulb has a lifetime between
100 and 400 hours?

Suppose that for a particular brand of
light bulb, the lifetime (in months) of any randomly selected bulb
follows an exponential distribution, with parameter l = 0.12
a) What are the mean and standard deviation for
the average lifetime of the particular brand of light bulbs? What
is the probability a single bulb will last greater than 9
months?
b)
If we randomly select 25 light bulbs of the particular
brand, what are the mean and standard...

17. The life of light bulbs is distributed normally. The
variance of the lifetime is 225 and the mean lifetime of a bulb is
590 hours. Find the probability of a bulb lasting for at most 600
hours. Round your answer to four decimal places.
18. Find the area under the standard normal curve to the left of
z=2.63. Round your answer to four decimal places, if necessary.

The lifetime of a lightbulb follows a normal distribution with
mean 1500 hours and standard deviation of 100 hours. a. What is the
probability that a lightbulb will last at least 1400 hours? b. What
is the probability that a light bulb burns out in fewer than 1600
hours? c. What is the probability that a light bulb burns out in
fewer than 1600 hours given that it has lasted 1400 hours? d. A
technology breakthrough has occurred for which...

problem 7. Lightbulbs of a certain type are advertised as having
an average lifetime of 750 hours. The price of these bulbs is very
favorable, so a potential customer has decided to go ahead with a
purchase arrangement unless it can be conclusively demonstrated
that the true average lifetime is smaller than what is advertised.
A random sample of 50 bulbs was selected, the lifetime of each bulb
determined, and the appropriate hypotheses were tested using
Minitab, resulting in the...

Problem 0.1
Suppose X and Y are two independent exponential random variables
with respective densities given by(λ,θ>0) f(x) =λe^(−xλ) for
x>0 and g(y) =θe^(−yθ) for y>0.
(a) Show that Pr(X<Y) =∫f(x){1−G(x)}dx {x=0, infinity] where
G(x) is the cdf of Y, evaluated at x [that is,G(x) =P(Y≤x)].
(b) Using the result from part (a), show that P(X<Y)
=λ/(θ+λ).
(c) You install two light bulbs at the same time, a 60 watt bulb
and a 100 watt bulb. The lifetime of the...

Suppose that the lifetime of a battery (in thousands of hours)
is a random variable X whose p.d.f. is given by:
{ 0 x <= 0
f(x) = {
{ ke^(-2x) 0 < x
(a) Find the constant k to make this a legitimate p.d.f.
Also sketch this p.d.f. [Hint: improper integral.]
(b) Determine the c.d.f. of X. Also sketch this c.d.f.
(c) Determine the mean [Hint: Integration By Parts] ...
... and median of X, and comment on how...

The life duration for a light bulb is well approximated by an
exponential random variable with a mean of 400 hours. Assume that a
classroom has a projector that has a life bulb with a lifetime
distribution as above, and it is used 40 hours per week. Also, for
your calculations, assume that a month has exactly 4 weeks.
What is the probability that you would need to replace the bulb
no more than twice a year? Assume uniform usage...

Q1 Electric cars of the same type have ranges,
X, that are normally distributed with a mean of 340 km and a
standard deviation of 20 km, when driven on a test track.
Define
(a) Find L and U such that
(b) Find the probability that the range on the test
track of a randomly chosen car of this type is between 300 km and
350 km.
(c) Given that , what range on the test track can the
manufacturer...

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