High School

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Assume that women's weights are normally distributed with a mean given by [tex]\mu = 143 \text{ lb}[/tex] and a standard deviation given by [tex]\sigma = 29 \text{ lb}[/tex].

(a) If 1 woman is randomly selected, find the probability that her weight is between 108 lb and 177 lb.

Answer :

Final answer:

To find the probability of a randomly selected woman's weight being between 108 lb and 177 lb with a mean of 143 lb and a standard deviation of 29 lb, one must calculate the z-scores for each weight and find the area under the standard normal curve corresponding to those z-scores.

Explanation:

The question involves finding the probability that a randomly selected woman's weight is between 108 lb and 177 lb, given that women's weights are normally distributed with a mean (μ) of 143 lb and a standard deviation (σ) of 29 lb.

We can solve this problem by calculating the z-scores for 108 lb and 177 lb and then using the standard normal distribution to find the probabilities corresponding to these z-scores. The z-score is calculated using the formula z = (X - μ) / σ, where X is the value for which we want to find the z-score.

  1. Calculate the z-score for 108 lb: z = (108 - 143) / 29
  2. Calculate the z-score for 177 lb: z = (177 - 143) / 29
  3. Use the standard normal distribution table or a calculator to find the area under the curve between these two z-scores.
  4. The result will give us the probability of a randomly selected woman's weight falling between 108 lb and 177 lb.

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