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Chapter 6: Q4E (page 370)

Suppose that a random sample of size n is to be taken from a distribution for which the mean is 渭, 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:

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

Short Answer

Expert verified

The smallest possible value of n is 385

Step by step solution

01

Given information

A random sample of size n taken from the distribution with a mean of \(\mu \) and standard deviation \(\sigma = 3\)

02

Finding the smallest value of n

The distribution,

\(\begin{array}{c}Z = \frac{{\sqrt n \left( {{{\bar X}_n} - \mu } \right)}}{\sigma }\\ = \frac{{\sqrt n \left( {{{\bar X}_n} - \mu } \right)}}{3}\end{array}\)

Then the distribution of Z will be an approximately standard normal distribution.

Therefore,

\(\begin{array}{c}\Pr \left( {\left| {{{\bar X}_n} - \mu } \right| < 0.3} \right) = \Pr \left( {\left| Z \right| < 0.1\sqrt n } \right)\\ \approx 2\Phi \left( {0.1\sqrt n } \right) - 1\end{array}\)

But,

\(\begin{array}{c}2\Phi \left( {0.1\sqrt n } \right) - 1 \ge 0.95\\2\Phi \left( {0.1\sqrt n } \right) \ge 1 + 0.95\\\Phi \left( {0.1\sqrt n } \right) \ge \frac{{1.95}}{2}\\0.1\sqrt n \ge {\Phi ^{ - 1}}\left( {0.975} \right)\end{array}\)

\(\begin{array}{c}0.1\sqrt n \ge 1.96\\\sqrt n \ge 19.6\\n \ge 384.16\end{array}\)

Therefore, the smallest possible value of n is 385

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Most popular questions from this chapter

Suppose that three girls, A, B, and C throw snowballs at a target. Suppose also that girl A throws 10 times, and the probability that she will hit the target on any given throw is 0.3; girl B throws 15 times, and the probability that she will hit the target on any given throw is 0.2, and girl C throws 20 times, and the probability that she will hit the target on any given throw is 0.1. Determine the probability that the target will be hit at least 12 times

Suppose that \({X_1},...,{X_n}\)form a random sample of size n from a distribution for which the mean is 6.5 and the variance is 4. Determine how large the value of n must be in order for the following relation to be satisfied:

\({\bf{P}}\left( {{\bf{6}} \le {{{\bf{\bar X}}}_{\bf{n}}} \le {\bf{7}}} \right) \ge {\bf{0}}{\bf{.8}}\)

Suppose that 16 digits are chosen at random with replacement from the set {0,...,9}. What is the probability that their average will lie between 4 and 6?

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\({{\bf{\sigma }}^{\bf{2}}}\).

  1. Determine the asymptotic distribution of the statistic

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

  1. Find a variance stabilizing transformation for the statistic

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