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ruslelena [56]
2 years ago
11

A local agricultural cooperative claims that 55% of about 60,000 adults in a country believe that gardening should be part of th

e school curriculum. What is the probability that, from a simple random sample of 300 adults in the county, less than 50% would say they believe that gardening should be part of the school curriculum
Mathematics
1 answer:
insens350 [35]2 years ago
8 0

Using the <u>normal distribution and the central limit theorem</u>, it is found that there is a 0.0409 = 4.09% probability that, from a simple random sample of 300 adults in the county, less than 50% would say they believe that gardening should be part of the school curriculum.

In a normal distribution with mean \mu and standard deviation \sigma, the z-score of a measure X is given by:

Z = \frac{X - \mu}{\sigma}

  • It measures how many standard deviations the measure is from the mean.  
  • After finding the z-score, we look at the z-score table and find the p-value associated with this z-score, which is the percentile of X.
  • By the Central Limit Theorem, the sampling distribution of sample proportions for a proportion p in a sample of size n has \mu = p, s = \sqrt{\frac{p(1 - p)}{n}}

In this problem:

  • The proportion is of 55%, hence p = 0.55
  • The sample has 300 adults, hence n = 300

Then, the <u>mean and the standard error</u> are given by:

\mu = p = 0.55

s = \sqrt{\frac{p(1 - p)}{n}} = \sqrt{\frac{0.55(0.45)}{300}} = 0.0287

The probability is the <u>p-value of Z when X = 0.5,</u> hence:

Z = \frac{X - \mu}{\sigma}

By the Central Limit Theorem

Z = \frac{X - \mu}{s}

Z = \frac{0.5 - 0.55}{0.0287}

Z = -1.74

Z = -1.74 has a p-value of 0.0409.

0.0409 = 4.09% probability that, from a simple random sample of 300 adults in the county, less than 50% would say they believe that gardening should be part of the school curriculum.

A similar problem is given at brainly.com/question/25800303

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