91Ó°ÊÓ

Given a random sample of size 24 from a normal distribution, find \(k\) such that (a) \(\mathrm{P}(-2.069

A normal population with unknown variance has a mean of \(20 .\) Is one likely to obtain a random sample of size \(!\) ) from this population with a mean of 24 and a standard deviation of \(4.1 ?\) If not, what conclusion would you draw?

A maker of a certain brand of low-fat cereal bars claims that their average saturated fat content is 0.5 gram. In a random sample of 8 cereal bars of this brand the saturated fat. content was \(0.6 .0 .7 .0 .7 .0 .3 .0 .4 .0 .5,\) \(0.4,\) and \(0.2 .\) Would you agree with the claim? Assume a normal distribution.

Pull-strength tests on 10 soldered leads for a semi conductor device yield the following results in pounds force required to rupture the bond: $$\begin{array}{llllll}19.8 & 12.7 & 13.2 & 16.9 & 10.6 \\ 18.8 & 11.1 & 143 & 17.0 & 12.5\end{array}$$ Another set of 8 leads was tested after encapsulation to determine whether the pull strength has been increased by encapsulation of the device, with the following results: $$\begin{array}{llllllll}24.9 & 22.8 & 23.6 & 22.1 & 20.4 & 21.6 & 21.8 & 22.5\end{array}$$ Comment on the evidence available concerning equality of the two population variances.

If the number of hurricanes that hit. a certain area of the eastern United States per year is a random variable having a Poisson distribution with \(p=6,\) find the probability that this area will be hit by (a) exactly 15 hurricanes in 2 years; (b) at most 9 hurricanes in 2 years.

The concentration of an active ingredient in the output of a chemical reaction is strongly influenced by the catalyst that is used in the reaction. It is felt that when catalyst \(A\) is used, the population mean concentration exceeds \(65 \%\). The standard deviation is known to be \(a=5 \%\) A sample of outputs from 30 independent experiments gives the average concentration of \(x_{A}=64.5 \%\). (a) Does this sample information with an average concentration of \(\bar{x}_{A}=64.5 \%\) provide disturbing information that perhaps \(\mu_{A}\) is not \(65 \%,\) but less than \(65 \% ?\) Support your answer with a probability statement. (b) Suppose a similar experiment is done with the use of another catalyst, catalyst \(B\). The standard deviation \(a\) is still assumed to be \(5 \%\) and \(\bar{x}_{B}\) turns out to be \(70 \%\). Comment on whether or not the sample information on catalyst \(B\) seems to give strong information that suggests that \(\mu_{B}\) is truly greater than \(\mu_{A}\). Support your answer by computing $$P\left(\bar{X}_{B}-X_{A} \geq 5.5 \quad \mid \mu_{B}=\mu_{A}\right)$$. (c) Under the condition that \(\mu_{A}=\mu_{B}=65 \%,\) give the approximate distribution of the following quantities (with mean and variance of each). Make use of the central limit theorem.

Suppose a filling machine is used to fill cartons with a liquid product. The specification that is strictly enforced for the filling machine is \(9 \pm 1.5\) oz. If any carton is produced with weight outside these bounds, it is considered by the supplier to be a defective. It is hoped that at least \(99 \%\) of cartons will meet these specifications. With the conditions \(\mu=9\) and \(a=1\), what proportion of cartons from the process are defective? If changes are made to reduce variability, what must \(a\) be reduced to in order to meet specifications with probability \(0.99 ?\) Assume a normal distribution for the weight.