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Exercises 5.6-1. Let X be the mean of a random sample of size 12 from the uniform distribution on the interval (0, 1). Approximate P(1/2 < X < 2/3).

Exercises 5.6-1. Let X be the mean of a random sample of size 12 from the uniform distribution on the interval (0, 1). Approximate P(1/2 < X < 2/3).

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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5.6-1

### Exercises **5.6-1.** Let \(\overline{X}\) be the mean of a random sample of size 12 from the uniform distribution on the interval (0, 1). Approximate \(P(1/2 \leq \overline{X} \leq 2/3)\). **5.6-2.** Let \(Y = X_1 + X_2 + \cdots + X_{15}\) be the sum of a random sample of size 15 from the distribution whose pdf is \(f(x) = (3/2)x^2, -1 < x < 1\). Using the pdf of \(Y\), we find that \(P(-0.3 \leq Y \leq 1.5) = 0.22788\). Use the central limit theorem to approximate this probability. **5.6-3.** Let \(\overline{X}\) be the mean of a random sample of size 36 from an exponential distribution with mean 3. Approximate \(P(2.5 \leq \overline{X} \leq 4)\). **5.6-4.** Approximate \(P(39.75 \leq \overline{X} \leq 41.25)\), where \(\overline{X}\) is the mean of a random sample of size 32 from a distribution with mean \(\mu = 40\) and variance \(\sigma^2 = 8\).
Transcribed Image Text:### Exercises **5.6-1.** Let \(\overline{X}\) be the mean of a random sample of size 12 from the uniform distribution on the interval (0, 1). Approximate \(P(1/2 \leq \overline{X} \leq 2/3)\). **5.6-2.** Let \(Y = X_1 + X_2 + \cdots + X_{15}\) be the sum of a random sample of size 15 from the distribution whose pdf is \(f(x) = (3/2)x^2, -1 < x < 1\). Using the pdf of \(Y\), we find that \(P(-0.3 \leq Y \leq 1.5) = 0.22788\). Use the central limit theorem to approximate this probability. **5.6-3.** Let \(\overline{X}\) be the mean of a random sample of size 36 from an exponential distribution with mean 3. Approximate \(P(2.5 \leq \overline{X} \leq 4)\). **5.6-4.** Approximate \(P(39.75 \leq \overline{X} \leq 41.25)\), where \(\overline{X}\) is the mean of a random sample of size 32 from a distribution with mean \(\mu = 40\) and variance \(\sigma^2 = 8\).
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