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A boat capsized and sank in the lake. The boat was rated to carry 70 passengers based on an assumed mean weight of 143 lb, resulting in a load limit of 10,010 lb. After the boat sank, the assumed mean weight for similar boats was changed from 143 lb to 171 lb.

Answer :

The final answer is the probability that the boat is overloaded because the mean weight of the 70 passengers is greater than 143lb is 1.0000.

A normal distribution is a continuous data distribution with a bell-shaped curve. The normally distributed random variable X has mean [tex]$\mu$[/tex] and standard deviation [tex]$\sigma$[/tex].

In addition, the standard normal distribution represents a normal curve with a mean of zero and a standard deviation of one.

The number of standard deviations an element deviates from the mean is indicated by a standardized Z-score.

What is the theorem of the central limit?

  • If the sample size from the population is sufficiently large and has a finite variance, the mean of all samples taken will be approximately the same as the population mean. And the variance would be the population variance divided by the sample size. In addition, the sampling distribution would be roughly normal.
  • The formula for sampling distribution of the sample mean is [tex]$\mu_{\bar{x}}=\mu$[/tex]
  • The formula for standard deviation of the sample mean is,[tex]\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}[/tex]
  • Here, [tex]$\sigma$[/tex] is the population standard deviation and n is the sample size.
  • The formula for standard z-score is,

[tex]z=\frac{\bar{x}-\mu_{x}}{\frac{\sigma}{\sqrt{n}}}[/tex]

  • From the given information,Population mean, [tex]$\mu=171$[/tex],Population standard deviation, [tex]$\sigma=37.7$[/tex],Sample size, n=70
  • Compute the probability that the boat is overloaded because the mean weight of the 70 passengers is greater than [tex]$143l \mathrm{~b}$[/tex].

[tex]$$\begin{aligned}P(\bar{X} > 143) &=P\left(\frac{\bar{X}-\mu}{\sigma / \sqrt{n}} > \frac{143-171}{37.7 / \sqrt{70}}\right) \\&=P(Z > -6.21) \\&=1-P(Z < -6.21) \text { (Use the Excel function NORMSDIST(-6.21)) } \\&=1-0.0000 \\&=1.0000\end{aligned}$$[/tex]

To learn more about standard deviation visit:

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