Questions: In the US, the weight of a randomly selected adult male is normally distributed with a mean of 195 lbs and a standard deviation of 12 lbs. Let X be the weight of a randomly selected adult male and let S be the total weight of a random sample of size 12. 1. Describe the probability distribution of X and state its parameters μ and σ : X ~ Select an answer (μ=, σ=) and find the probability that the weight of a randomly selected adult male is more than 196 lbs. (Round the answer to 4 decimal places) 2. Use the Central Limit Theorem Select an answer to describe the probability distribution of S and state its parameters μS and σS : (Round the answers to 1 decimal place) S ~ Select an answer (μS=, σS=) and find the probability that the total weight of a sample of 12 randomly selected adult males is between 2354 and 2365 lbs. (Round the answer to 4 decimal places)

In the US, the weight of a randomly selected adult male is normally distributed with a mean of 195 lbs and a standard deviation of 12 lbs. Let X be the weight of a randomly selected adult male and let S be the total weight of a random sample of size 12. 1. Describe the probability distribution of X and state its parameters μ and σ : X ~ Select an answer (μ=, σ=) and find the probability that the weight of a randomly selected adult male is more than 196 lbs. (Round the answer to 4 decimal places) 2. Use the Central Limit Theorem Select an answer to describe the probability distribution of S and state its parameters μS and σS : (Round the answers to 1 decimal place) S ~ Select an answer (μS=, σS=) and find the probability that the total weight of a sample of 12 randomly selected adult males is between 2354 and 2365 lbs. (Round the answer to 4 decimal places)
Transcript text: In the US, the weight of a randomly selected adult male is normally distributed with a mean of 195 lbs and a standard deviation of 12 lbs . Let $X$ be the weight of a randomly selected adult male and let $S$ be the total weight of a random sample of size 12. 1. Describe the probability distribution of $X$ and state its parameters $\mu$ and $\sigma$ : $X \sim$ Select an answer $\boldsymbol{\theta}(\mu=$ $\square$ $\square$ $\sigma=$ ) and find the probability that the weight of a randomly selected adult male is more than 196 lbs. $\square$ (Round the answer to 4 decimal places) 2. Use the Central Limit Theorem Select an answer to describe the probability distribution of $S$ and state its parameters $\mu_{S}$ and $\sigma_{S}$ : (Round the answers to 1 decimal place) $S \sim$ Select an answer $0\left(\mu_{S}=\right.$ $\square$ , $\sigma_{S}=$ $\square$ ) and find the probability that the total weight of a sample of 12 randomly selected adult males is between 2354 and 2365 lbs. $\square$ (Round the answer to 4 decimal places) Question Help: Video Written Example Message instructor D Post to forum
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Solution

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Solution Steps

Step 1: Probability Distribution of \( X \)

The weight of a randomly selected adult male, \( X \), is normally distributed as follows: \[ X \sim N(195, 12) \] where \( \mu = 195 \) lbs and \( \sigma = 12 \) lbs.

Step 2: Probability that \( X > 196 \)

To find the probability that the weight of a randomly selected adult male is more than 196 lbs, we calculate: \[ P(X > 196) = 1 - P(X \leq 196) = \Phi(\infty) - \Phi(0.0833) = 0.4668 \] Thus, the probability that \( X > 196 \) is: \[ \boxed{0.4668} \]

Step 3: Probability Distribution of \( S \)

Using the Central Limit Theorem, the total weight of a sample of size 12, \( S \), is normally distributed as follows: \[ S \sim N(2340, 41.5692) \] where \( \mu_S = 195 \times 12 = 2340 \) lbs and \( \sigma_S = 12 \times \sqrt{12} \approx 41.5692 \) lbs.

Step 4: Probability that \( 2354 < S < 2365 \)

To find the probability that the total weight of a sample of 12 randomly selected adult males is between 2354 and 2365 lbs, we calculate: \[ P(2354 < S < 2365) = \Phi(2.0833) - \Phi(1.1667) = 0.1031 \] Thus, the probability that \( 2354 < S < 2365 \) is: \[ \boxed{0.1031} \]

Final Answer

  • Probability that \( X > 196 \): \( \boxed{0.4668} \)
  • Probability that \( 2354 < S < 2365 \): \( \boxed{0.1031} \)
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