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Define suitable populations from which the following samples are selected: (a) Persons in 200 homes are called by telephone in the city of Richmond and asked to name the candidate that they favor for election to the school board. (b) A coin is tossed 100 times and 34 tails are recorded. (c) Two hundred pairs of a new type of tennis shoe were tested on the professional tour and, on the average, lasted 4 months. (d) On five different occasions it took a lawyer \(21,26,\) \(24,22,\) and 21 minutes to drive from her suburban home to her midtown office.

The numbers of incorrect answers on a true-false competency test for a random sample of 15 students were recorded as follows: 2,1,3,0,1,3,6,0,3,3,5 , \(2,1,4,\) and \(2 .\) Find (a) the mean; (b) the median; (c) the mode.

The lengths of time, in minutes, that 10 patients waited in a doctor's office before receiving treatment were recorded as follows: \(5,11,9,5,10,15,6,10,5,\) and \(10 .\) Treating the data as a random sample, find (a) the mean; (b) the median; (c) the mode.

The reaction times for a random sample of 9 subjects to a stimulant were recorded as \(2.5,3.6,3.1,4.3,\) 2.9. \(2.3,2.6,4.1,\) and 3.4 seconds. Calculate (a) the mean; (b) the median.

The tar contents of 8 brands of cigarettes selected at random from the latest list released by the Federal Trade Commission are as follows: 7.3,8.6,10.4 \(16.1,12.2,15.1,14.5,\) and 9.3 milligrams. Calculate (a) the mean; (b) the variance.

(a) Show that the sample variance is unchanged if a constant \(\mathrm{c}\) is added to or subtracted from each value in the sample. (b) Show that the sample variance becomes \(\mathrm{c}^{2}\) times its original value if each observation in the sample is multiplied by \(\mathrm{c}\).

If all possible samples of size 16 are drawn from a normal population with mean equal to 50 and standard deviation equal to \(5,\) what is the probability that a sample mean \(\bar{X}\) will fall in the interval from \(\mu_{\bar{X}}-1.9 \sigma_{\bar{X}}\) to \(\mu_{X} \sim 0.4 \sigma_{\bar{X}}\) ? Assume that the sample means can be measured to any degree of accuracy.

Given the discrete uniform population $$ I(x)=\left\\{\begin{array}{ll}\frac{1}{3}, & x=2,4,6 \\\0, & \text { }\end{array}\right.$$ find the probability that a random sample of size 54 selected with replacement, will yield a sample mean greater than 4.1 but less than 4.4 . Assume the means to be measured to the nearest tenth.

A certain type of thread is manufactured with a mean tensile strength of 78.3 kilograms and a standard deviation of 5.6 kilograms. How is the variance of the sample mean changed when the sample size is (a) increased from 64 to \(196 ?\) (b) decreased from 784 to \(49 ?\)

If the standard deviation of the mean for the sampling distribution of random samples of size 36 from a large or infinite population is \(2,\) how large must the size of the sample become if the standard deviation is to be reduced to \(1.2 ?\)