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The heights of 1000 students are approximately normally distributed with a mean of 174.5 centimeters and a standard deviation of 6.9 centimeters. If 200 random samples of size 25 are drawn from this population and the means recorded to the nearest tenth of a centimeter, determine (a) the mean and standard deviation of the sampling distribution of \(\bar{X}\); (b) the number of sample means that fall between 172.5 and 175.8 centimeters inclusive; (c) the number of sample means falling below 172.0 centimeters.

The random variable \(X\), representing the number of cherries in a cherry puff, has the following probability distribution $$\begin{array}{c|cccc} x & 4 & 5 & 6 & 7 \\\\\hline P(X=x) & 0.2 & 0.4 & 0.3 & 0.1\end{array}$$ (a) Find the mean \(p\). and the variance \(a^{2}\) of \(X\). (b) Find the mean \(\mu_{\bar{X}},\) and the variance \(\sigma_{\bar{X}}^{2}\) of the mean \(\bar{X}\) for random samples of 36 cherry puffs. (c) Find the probability that the average number of cherries in 36 cherry puffs will be less than 5.5 .

The amount of time that a drive-through bank teller spends on a customer is a random variable with a mean \(\mu=3.2\) minutes and a standard deviation \(\sigma=1.6\) minutes. If a random sample of 64 customers is observed, find the probability that their mean time at the teller's counter is (a) at most 2.7 minutes: (b) more than 3.5 minutes; (c) at least 3.2 minutes but less than 3.4 minutes.

In a chemical process the amount of a certain type of impurity in the output is difficult to control and is thus a random variable. Speculation is that the population mean amount of the impurity is 0.20 grams per gram of output. It is known that the standard deviation is 0.1 grams per gram. An experiment is conducted to gain more insight regarding the speculation that \(\mu=0.2\). The process was run on a lab scale 50 times and the sample average \(x\) turned out to be 0.23 grams per gram. Comment on the speculation that the mean amount of impurity is 0.20 grams per gram. Make use of the central limit theorem in your work.

A random sample of size 25 is taken from a normal population having a mean of 80 and a standard deviation of \(5 .\) A second random sample of size 36 is taken from a different normal population having a mean of 75 and a standard deviation of 3 . Find the probability that the sample mean computed from the 25 measurements will exceed the sample mean computed from the 36 measurements by at least 3.4 but less than 5.9 . Assume the difference of the means to be measured to the nearest tenth.

The distribution of heights of a certain breed of terrier dogs has a mean height of 72 centimeters and a standard deviation of 10 centimeters, whereas the distribution of heights of a certain breed of poodles has a mean height of 28 centimeters with a standard deviation of 5 centimeters. Assuming that the sample means can be measured to any degree of accuracy, find the probability that the sample mean for a random sample of heights of 64 terriers exceeds the sample mean for a random sample of heights of 100 poodles by at most 44.2 centimeters.

The chemical benzene is highly toxic to humans. However, it is used in the manufacture of many medicine dyes, leather, and many coverings. In any production process involving benzene, the water in the output of the process must not exceed 7950 parts per million (ppm) of benzene because of government regulations. For a particular process of concern the water sample was collected by a manufacturer 25 times randomly and the sample average \(x\) was \(7960 \mathrm{ppm}\). It is known from historical data that the standard deviation \(a\) is \(100 \mathrm{ppm}\) (a) What is the probability that the sample average in this experiment would exceed the government limit if the population mean is equal to the limit? Use the central limit theorem. (b) Is an observed \(\bar{x}=7960\) in this experiment firm evidence that the population mean for the process exceeds the government limit? Answer your question by computing $$P\left(X^{-}>_{-} 7960 \mid p=7950\right).$$ Assume that the distribution of benzene concentration is normal.

The scores on a placement test given to college freshmen for the past five years are approximately normally distributed with a mean \(\mu=71\) and a variance \(a^{2}=8\). Would you still consider \(\sigma^{2}=8\) to be a valid value of the variance if a random sample of 20 students who take this placement test this year obtain a value of \(s^{2}=20 ?\)

Show that the variance of \(S^{2}\) for random samples of size \(n\) from a normal population decreases as \(n\) becomes large. [Hint: First, find the variance of \((\mathrm{n}-1) S^{2} / \sigma^{2}\).

(a) Find \(t_{0.025}\) when \(v=14\). (b) Find \(-t_{0.10}\) when \(v=10\). (c) Find \(t_{0.995}\) when \(v=7\).