Department of Agriculture (USDA) found that the proportion of y st is 0.238. Suppose that Lance, a nutritionist, surveys the dietary adults ages 20-39 in the United States. imit theorem to find the probability that the number of individuals, reater than 122. You may find table of critical values helpful. as a decimal precise to three places.
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
![**The Central Limit Theorem and Young Adults Skipping Breakfast**
The United States Department of Agriculture (USDA) found that the proportion of young adults ages 20-39 who regularly skip eating breakfast is 0.238. Suppose that Lance, a nutritionist, surveys the dietary habits of a random sample of size \( n = 500 \) of young adults ages 20-39 in the United States.
**Problem Statement:**
Apply the central limit theorem to find the probability that the number of individuals, \( X \), in Lance's sample who regularly skip breakfast is greater than 122. You may find a table of critical values helpful.
Express the result as a decimal precise to three places.
\[ P(X > 122) = \]
**Explanation of the Central Limit Theorem (for Educational Purposes):**
The central limit theorem states that the sampling distribution of the sample mean (or the sum) of a large number of independent, identically distributed variables will be approximately normally distributed, regardless of the original distribution of the variables. This theorem helps in estimating probabilities and is a cornerstone of statistics, especially in inferential statistics.
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By implementing this theorem, statisticians can approximate normal distribution for sample sizes typically 30 or more, enabling the application of normal probability tools to solve real-world problems like Lance's study.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2fb7a5f7-d1fe-47c3-ab51-c010c8f81dff%2F45aa7ac2-8d91-4127-b3b5-166c4f87c953%2Fxjmf38_processed.jpeg&w=3840&q=75)
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