We appreciate your visit to Application of Normal Distribution Finding the Probability and Scores The weights of adults living in the town of Metaluna are normally distributed with a mean. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!
Answer :
The probability that an adult sampled at random will weigh between 136 pounds and 164 pounds can be calculated using the properties of the normal distribution.
To find the probability, we need to calculate the area under the normal curve between the two weight values. We can convert the given weights into z-scores (standardized scores) using the formula:
z = (x - μ) / σ
where x is the given weight, μ is the mean weight, and σ is the standard deviation.
For the lower weight value of 136 pounds:
z1 = (136 - 146) / 12.7 = -0.79
For the upper weight value of 164 pounds:
z2 = (164 - 146) / 12.7 = 1.42
Now, we can look up the corresponding z-scores in the standard normal distribution table or use a calculator to find the area under the curve between these z-scores. The probability is equal to the difference between these two areas.
Using a standard normal distribution table or calculator, we find the area to the left of z1 (0.2139) and the area to the left of z2 (0.9236). Therefore, the probability of an adult weighing between 136 and 164 pounds is:
P(136 < x < 164) = P(-0.79 < z < 1.42) = P(z < 1.42) - P(z < -0.79) = 0.9236 - 0.2139 = 0.7097
The probability that an adult sampled at random will weigh between 136 and 164 pounds is approximately 0.7097 or 70.97%. This means that there is a 70.97% chance of randomly selecting an adult whose weight falls within this range in the town of Metaluna, assuming the weights follow a normal distribution with a mean of 146 pounds and a standard deviation of 12.7 pounds.
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