Respuesta :
Answer:
Step-by-step explanation:
To find the probability that the capacity will be exceeded with 16 random people in the elevator, given that weights are normally distributed with a mean of 170.5 pounds and a standard deviation of 25.2 pounds, we'll use the concept of the Central Limit Theorem for sample means.
The Central Limit Theorem states that for a large enough sample size (in this case, 16), the distribution of the sample means will be approximately normal, regardless of the distribution of the individual data points.
First, let's calculate the standard error of the mean (SE), which is given by the formula:
\[ SE = \frac{\text{Standard Deviation}}{\sqrt{\text{Sample Size}}} \]
Plugging in the values:
\[ SE = \frac{25.2}{\sqrt{16}} = \frac{25.2}{4} = 6.3 \text{ pounds} \]
Next, we need to find the z-score for a mean weight of 160 pounds using the formula:
\[ z = \frac{\text{Sample Mean} - \text{Population Mean}}{\text{SE}} \]
\[ z = \frac{160 - 170.5}{6.3} = \frac{-10.5}{6.3} \approx -1.67 \]
Now, we can use a standard normal distribution table or a calculator to find the probability corresponding to a z-score of -1.67. The probability (P) that the capacity will be exceeded is the area to the right of this z-score.
Using a standard normal distribution table or calculator, the probability is approximately 0.9525 (rounded to four decimal places).
Therefore, the probability that the capacity will be exceeded with 16 random people in the elevator is approximately 0.9525 or 95.25%.
