Question

Suppose the breaking strength of plastic bags is a Gaussian random variable. Bags from company 1 have a mean strength of 8 kilograms and a variance of 1 kg2 ; Bags from company 2 have a mean strength of 9 kilograms and a variance of 0.5 kg2 . Assume we check the sample mean ?̅ 10 of the breaking strength of 10 bags, and use ?̅ 10 to determine whether a batch of bags comes from company 1 (null hypothesis ?0) or from company 2 (alternative hypothesis ?1). The decision rule is: Accept ?0, if ?̅ 10 ≤ ?; Reject ?0 and accept ?1, if ?̅ 10 > ?. a) (10 pts) Find the threshold ? so that the probability of type I error ? is 5%. b) (5 pts) Calculate ?, the probability of type II error, under this test.

Answer #1

The Tough Twine Company claims that their twine has an average
breaking strength of 16.5 pounds. The head of the shipping
department at Korber’s hardware store suspects that this figure is
too high and wishes to test the appropriate hypothesis. He takes a
random sample of 36 pieces and finds that the mean breaking
strength of the sample is 15.8 lbs with s = 2.9 lbs.
(a) State the null hypothesis and alternative hypothesis
(b) Find the test statistic
(c)...

Macarthurs, a manufacturer of ropes used in abseiling, wished to
determine if the production of their ropes was performing according
to their specifications. All ropes being manufactured were required
to have an average breaking strength of 228.5 kilograms and a
standard deviation of 27.3 kilograms. They planned to test the
breaking strength of their ropes using a random sample of forty
ropes and were prepared to accept a Type I error probability of
0.01.
1. State the direction of the...

Let Θ be a Bernoulli random variable that indicates which one of
two hypotheses is true, and let P(Θ=1)=p.
Under the hypothesis Θ=0, the random variable X has a
normal distribution with mean 0, and variance 1. Under the
alternative hypothesis Θ=1, X has a normal distribution
with mean 2 and variance 1.
Suppose for this part of the problem that p=2/3. The MAP
rule can choose in favor of the hypothesis Θ=1 if and only if
x≥c1. Find the...

For a random sample of 50 measurements of the breaking strength
of Brand A cotton threads, sample mean ¯x1 = 210 grams,
sample standard deviation s1 = 8 grams. For Brand B,
from a random sample of 50, sample mean ¯x2 = 200 grams
and sample standard deviation s2 = 25 grams. Assume that
population distributions are approximately normal with unequal
variances. Answer the following questions 1 through 3.
1. What is the (estimated) standard error of difference between
two...

1. The breaking strengths (measured in dynes) of nylon fibers
are normally distributed with a
mean of 12,500 and a variance of 202,500.
a) What is the probability that a fiber strength is more than
13,175?
b) What is the probability that a fiber strength is less than
11,600?
c) What is the probability that a fiber strength is between 12,284
and 15,200?
d) What is the 90 th percentile of the fiber breaking strength?
2.
Suppose that X, Y...

1) Choose from the Following: Gaussian Distribution, Empirical
Rule, Standard Normal, Random Variable, Inverse Normal, Normal
Distribution, Approximation, Standardized, Left Skewed, or
Z-Score.
Same as the Normal Distribution
A statement whose reliability is based
upon observation and experimental evidence; formulas based upon
experience rather than mathematical conclusions.
A [_______ ________] is a quantity
resulting from a random experiment that can assume different values
by chance.
A result that is not exact, but is
accurate enough for some specific purposes.
The...

1) Suppose a random variable, x, arises from a binomial
experiment. Suppose n = 6, and p = 0.11.
Write the probability distribution. Round to six decimal places,
if necessary.
x
P(x)
0
1
2
3
4
5
6
Find the mean.
μ =
Find the variance.
σ2 =
Find the standard deviation. Round to four decimal places, if
necessary.
σ =
2) Suppose a random variable, x, arises from a binomial
experiment. Suppose n = 10, and p =...

DUrable press cotton fabrics are treated to improve their
recovery from wrinkles after washing. Unfortunately, the treatment
also reduces the strength of the fabric. The breaking strength of
untreated fabric is normally distributed with mean 52.3 pounds and
standard deviation 1.9 pounds. The same type of fabric after
treatment has normally distributed breaking strength with mean 27.6
pounds and standard deviation 1.6 pounds. A clothing manufacturer
tests 5 specimens of each fabric. All 10 strength measurements are
independent. .1. What...

Suppose that X has probability function
fX(x)=cx2 for
0<x<1.
(a) (5 pts) Find c.
(b) (5 pts) Compute the cdf, FX(x).
(c) (5 pts) Find P(-1 ≤ X ≤ 0.5) .
(d) (5 pts) Find the moment-generating function(mgf) of X.
(e) (10 pts) Use the mgf to find the values of (i) the mean and
(ii) the variance of X.

A coffee company sells bags of coffee beans with an advertised
weight of 454 grams. A random sample of 20 bags of coffee beans has
an average weight of 457 grams. Weights of coffee beans per bag are
known to follow a normal distribution with standard deviation 7
grams. (a) Construct a 95% confidence interval for the true mean
weight of all bags of coffee beans. (Instead of typing ±, simply
type +-.) (1 mark) (b) Provide an interpretation of...

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