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The weights of 1500 bodybuilders are normally distributed with a mean of 190.6 pounds and a standard deviation of 5.8 pounds.

A. About how many bodybuilders are between 180 and 190 pounds?

B. What is the probability that a bodybuilder selected at random has a weight greater than 195 pounds?

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semsee45

Answer:

A.  637

B.  0.2240 (4 dp) = 22.4% (nearest tenth)

Step-by-step explanation:

Normal Distribution

[tex]\sf X \sim N(\mu, \sigma^2)[/tex]

Given:

  • [tex]\sf \mu = 190.6[/tex]
  • [tex]\sf \sigma=5.8[/tex]

[tex]\implies \sf X \sim N(190.6, 5.8^2)[/tex]

Part A

[tex]\begin{aligned}\sf P(180 < X < 190) & = \sf P(X < 190)-P(X\leq 180)\\& = \sf 0.4588035995-0.03380583874\\& = \sf 0.4249977608\end{aligned}[/tex]

Total number of bodybuilders = 1500

Therefore, the number of bodybuilders between 180 and 190 pounds is:

[tex]\begin{aligned}\sf P(180 < X < 190) \cdot1500 & = \sf 0.4249977608 \cdot 1500\\& = \sf 637.4966412\\& = \sf 637\end{aligned}[/tex]

Part B

[tex]\begin{aligned}\sf P(X > 195) & = \sf 1-P(X\leq 195)\\& = \sf 1-0.7759602537\\ & = \sf 0.2240397463\\ & = \sf 0.2240\:(4\:dp)\end{aligned}[/tex]

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