91Ó°ÊÓ

A random sample of n items is to be taken from a distribution with mean μ and standard deviation σ.

a. Use the Chebyshev inequality to determine the smallest number of items n that must be taken to satisfy the following relation:

\({\bf{Pr}}\left( {\left| {{{{\bf{\bar X}}}_{\bf{n}}}{\bf{ - \mu }}} \right| \le \frac{{\bf{\sigma }}}{{\bf{4}}}} \right) \ge {\bf{0}}{\bf{.99}}\)

b. Use the central limit theorem to determine the smallest number of items n that must be taken to satisfy the relation in part (a) approximately

Let f be a p.f. for a discrete distribution. Suppose that\(f\left( x \right) = 0\)for \(x \notin \left[ {0,1} \right]\). Prove that the variance of this distribution is at most\(\frac{1}{4}\). Hint: Prove that there is a distribution supported on just the two points\(\left\{ {0,1} \right\}\)with variance at least as large as f, and then prove that the variance of distribution supported on\(\left\{ {0,1} \right\}\)is at most\(\frac{1}{4}\).

Suppose that\({{\bf{X}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{X}}_{\bf{n}}}\)form a random sample from a normal distribution with mean 0 and unknown variance\({{\bf{\sigma }}^{\bf{2}}}\).

\({\left( {\frac{{\bf{1}}}{{\bf{n}}}\sum\nolimits_{{\bf{i = 1}}}^{\bf{n}} {{{\bf{X}}^{\bf{2}}}_{\bf{i}}} } \right)^{{\bf{ - 1}}}}\).

\(\left( {\frac{{\bf{1}}}{{\bf{n}}}\sum\nolimits_{{\bf{i = 1}}}^{\bf{n}} {{{\bf{X}}^{\bf{2}}}_{\bf{i}}} } \right)\).

Let X have the binomial distribution with parameters n and p. Let Y have the binomial distribution with parameters n and p/k with k > 1. Let \(Z = kY\).

a. Show that X and Z have the same mean.

b. Find the variances of X and Z. Show that, if p is small, then the variance of Z is approximately k times as large as the variance of X.

c. Show why the results above explain the higher variability in the bar heights in Fig. 6.2 compared to Fig. 6.1.

Suppose that a pair of balanced dice are rolled 120 times, and let X denote the number of rolls on which the sum of the two numbers is 7. Use the central limit theorem to determine a value of k such that\({\rm P}\left( {\left| {X - 20} \right| \le k} \right)\)is approximately 0.95.

Let X denote the total number of successes in 15 Bernoulli trials, with a probability of success p=0.3 on each trial.

1. Determine approximately the value of\({\rm P}\left( {X = 4} \right)\)by using the central limit theorem with the correction for continuity.
2. Compare the answer obtained in part (a) with the exact value of this probability.

Suppose that 75 percent of the people in a certain metropolitan area live in the city and 25 percent of the people live in the suburbs. If 1200 people attending a certain concert represent a random sample from the metropolitan area, what is the probability that the number of people from the suburbs attending the concert will be fewer than 270?

Suppose that the distribution of the number of defects on any given bolt of cloth is the Poisson distribution with a mean 5, and the number of defects on each bolt is counted for a random sample of 125 bolts. Determine the probability that the average number of defects per bolt in the sample will be less than 5.5.

Suppose that a random sample of size n is to be taken from a distribution for which the mean is μ, and the standard deviation is 3. Use the central limit theorem to determine approximately the smallest value of n for which the following relation will be satisfied:

\({\bf{Pr}}\left( {\left| {{{{\bf{\bar X}}}_{\bf{n}}}{\bf{ - \mu }}} \right|{\bf{ < 0}}{\bf{.3}}} \right) \ge {\bf{0}}{\bf{.95}}\).

Let X be a random variable for which \({\bf{E}}\left( {\bf{X}} \right){\bf{ = \mu }}\)and\({\bf{Var}}\left( {\bf{X}} \right){\bf{ = }}{{\bf{\sigma }}^{\bf{2}}}\).Construct a probability distribution for X such that \({\bf{P}}\left( {\left| {{\bf{X - \mu }}} \right| \ge {\bf{3\sigma }}} \right){\bf{ = }}\frac{{\bf{1}}}{{\bf{9}}}\)