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5.5-4. Let X equal the weight of the soap in a "6-pound" box. Assume that the distribution of X is N(6.05, 0.0004). (a) Find P(X < 6.0171). (b) If nine boxes of soap are selected at random from the production line, find the probability that at most two boxes weigh less than 6.0171 pounds each. HINT: Let Y equal the number of boxes that weigh less than 6.0171 pounds.

5.5-4. Let X equal the weight of the soap in a "6-pound" box. Assume that the distribution of X is N(6.05, 0.0004). (a) Find P(X < 6.0171). (b) If nine boxes of soap are selected at random from the production line, find the probability that at most two boxes weigh less than 6.0171 pounds each. HINT: Let Y equal the number of boxes that weigh less than 6.0171 pounds.

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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5.5-4 b only

**Transcription of the Image** The content appears to be mathematical problems related to normal distribution and probability. 1. **Problem 5.5-4**: Let \( X \) equal the weight of the soap in a "6-pound" box. Assume that the distribution of \( X \) is \( N(6.05, 0.0004) \). - (a) Find \( P(X < 6.0171) \). - (b) If nine boxes of soap are selected at random from the production line, find the probability that at most two boxes weigh less than 6.0171 pounds each. *Hint: Let \( Y \) equal the number of boxes that weigh less than 6.0171 pounds.* - (c) Let \( \overline{X} \) be the sample mean of the nine boxes. Find \( P(\overline{X} \leq 6.035) \). 2. **Problem**: \( N(40.56, 4.096) \). Let \( \overline{X} \) be the sample mean of a random sample of \( n = 16 \) observations of \( X \). - (a) Give the values of \( E(\overline{X}) \) and \( \text{Var}(\overline{X}) \). - (b) Find \( P(44.42 \leq \overline{X} \leq 48.98) \). The problems involve calculating probabilities, expectations, variances, and using concepts from statistics regarding the normal distribution and sampling.
Transcribed Image Text:**Transcription of the Image** The content appears to be mathematical problems related to normal distribution and probability. 1. **Problem 5.5-4**: Let \( X \) equal the weight of the soap in a "6-pound" box. Assume that the distribution of \( X \) is \( N(6.05, 0.0004) \). - (a) Find \( P(X < 6.0171) \). - (b) If nine boxes of soap are selected at random from the production line, find the probability that at most two boxes weigh less than 6.0171 pounds each. *Hint: Let \( Y \) equal the number of boxes that weigh less than 6.0171 pounds.* - (c) Let \( \overline{X} \) be the sample mean of the nine boxes. Find \( P(\overline{X} \leq 6.035) \). 2. **Problem**: \( N(40.56, 4.096) \). Let \( \overline{X} \) be the sample mean of a random sample of \( n = 16 \) observations of \( X \). - (a) Give the values of \( E(\overline{X}) \) and \( \text{Var}(\overline{X}) \). - (b) Find \( P(44.42 \leq \overline{X} \leq 48.98) \). The problems involve calculating probabilities, expectations, variances, and using concepts from statistics regarding the normal distribution and sampling.
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