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Chapter 6: Q6E (page 359)

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:

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

Short Answer

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80

Step by step solution

01

Given information

Mean = 6.5

Variance = 4

We need to calculate the sample size such that \(P\left( {6 \le {{\bar X}_n} \le 7} \right) \ge 0.8\)

02

Calculation of sample size

Chebyshev inequality

Let X be the random variable for which \({\rm{Var}}\left( X \right)\) exists. Then every for every number \(t > 0\) we have the following inequality

\(P\left( {\left| {X - E\left( X \right)} \right| \ge t} \right) \le \frac{{{\rm{Var}}\left( X \right)}}{{{t^2}}}\)

Using Chebyshev inequality

\(P\left( {6 \le {{\bar X}_n} \le 7} \right) \ge 0.8\)

\(\begin{array}{c}P\left( {\left| {{{\bar X}_n} - \mu } \right| < \frac{1}{2}} \right) \ge 1 - \frac{4}{{n \times \frac{1}{4}}}\\ = 1 - \frac{{16}}{n}\end{array}\)

Now

\(\begin{array}{c}1 - \frac{{16}}{n} \ge 0.8\\n - 16 \ge 0.8n\\n \ge 80\end{array}\)

So, the required sample size is 80.

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Most popular questions from this chapter

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.

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 X is a random variable for which E(X) = 渭 and \({\bf{E}}\left[ {{{\left( {{\bf{X - \mu }}} \right)}^{\bf{4}}}} \right]{\bf{ = }}{{\bf{\beta }}^{\bf{4}}}\) Prove that

\({\bf{P}}\left( {\left| {{\bf{X - \mu }}} \right| \ge {\bf{t}}} \right) \le \frac{{{{\bf{\beta }}_{\bf{4}}}}}{{{{\bf{t}}^{\bf{4}}}}}\)

Let\({X_1},{X_2},....{X_{30}}\)be independent random variables each having a discrete distribution with p.f.

\(f\left( x \right) = \left\{ \begin{array}{l}\frac{1}{4}\;\;\;\;\;\;if\;\;x = 0\;or\;2\\\frac{1}{2}\;\;\;\;\;\;if\;\;x = 1\\0\;\;\;\;\;\;\;otherwise\end{array} \right.\)

Use the central limit theorem and the correction for continuity to approximate the probability that\({X_1} + \cdots + {X_{30}}\)is at most 33.

Suppose that ,, and \(g\left( {z,y} \right)\)is a function that is continuous at \(\left( {z,y} \right) = \left( {b,c} \right)\). Prove that \(g\left( {{Z_n},{Y_n}} \right)\)converges in probability to \(g\left( {b,c} \right)\).
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