Answer the following:

a) The temperature recorded by a certain thermometer when placed in boiling water (true temperature 100 ∘C) is normally distributed with mean [tex]\mu=99.8[/tex] ∘C and standard deviation [tex]\sigma=0.1[/tex] ∘C. What is the probability that the thermometer reading is within ±0.05 ∘C of the true temperature?

b) If bolt thread length is normally distributed, what is the probability that the thread length of a randomly selected bolt is farther than 2.5 standard deviations from its mean value?

c) A machine used to extract juice from oranges obtains an amount from each orange that is approximately normally distributed, with a mean of 4.70 ounces and a standard deviation of 0.40 ounces. Ninety percent of the oranges will contain an amount that is between what two values (symmetrically distributed around the mean)?

Exercise 2 (5 marks)

One design for a system requires the installation of two identical components. The system will work if at least one of the components works. An alternative design requires four of these components, and the system will work if at least two of the four components work. If the probability that a component works is 0.9, and if the components function independently, which design has the greater probability of functioning?

Exercise 3 (8 marks)

Prove that the variance of the exponential distribution is [tex]\frac{1}{\lambda^2}[/tex].

Exercise 4 (10 marks)

A theoretical justification based on a certain material failure mechanism underlies the assumption that ductile strength of a material has a lognormal distribution. Suppose the values of the parameters are [tex]\mu=5[/tex] and [tex]\sigma=0.1[/tex].

a) Find the mean strength.
b) Find the median strength.
c) What proportion of material specimens have a ductile strength between 110 and 130?
d) If the smallest 5% of strength values were unacceptable, what would be the minimum acceptable strength?

Exercise 5 (11 marks)

Suppose that the random variable [tex]x[/tex] has a Weibull distribution with parameters [tex]\alpha=2.3[/tex] and [tex]\lambda=1.7[/tex] (where [tex]\lambda=\beta[/tex]). Find:

a) The higher quartile of the distribution.
b) The probability less than 1.3 (exclusive).
c) A formula for [tex]P(x > c)[/tex].

The questions revolve around statistical analysis including probability distributions and computations relating to normal, binomial, exponential, lognormal, and Weibull distributions.

Explanation:

The calculations for these problems involve a variety of techniques, including probability calculations for normally distributed data, calculations for a bolt thread, normally distributed juice extraction, systems functioning, and variance of an exponential distribution. Probability distributions such as the Weibull and lognormal distribution are also required.

For the first part, a Z score should be calculated and using the statistical table, you find the probability. The second part also involves a Z score, but it also necessitates knowledge of the empirical rule (or 68-95-99.7 rule). The part with the orange juicer also necessitates a Z score calculation to determine the range containing 90% of values.

For the design comparison, the binomial distribution can be applied. Variance of exponential distribution can be proved using the integral calculus formula. Questions about strength of a material with lognormal distribution require knowledge of the properties of this distribution: (1) the mean and (2) the median are exponentiated means and medians of the log transformed data. The computation of quartile and probability in a Weibull distribution involves its cumulative distribution function.

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