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HACTEHA [7]
2 years ago
12

A large car dealership claims that 47% of its customers are looking to buy a sport utility vehicle (SUV). A random sample of 61

customers is surveyed. What is the probability that less than 40% of the sample are looking to buy an SUV?
Mathematics
1 answer:
svetlana [45]2 years ago
7 0

Using the <em>normal distribution and the central limit theorem</em>, it is found that the probability is of 0.1368 = 13.68%.

<h3>Normal Probability Distribution</h3>

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, for a proportion p in a sample of size n, the sampling distribution of sample proportion is approximately normal with mean \mu = p and standard deviation s = \sqrt{\frac{p(1 - p)}{n}}, as long as np \geq 10 and n(1 - p) \geq 10.

In this problem:

  • 47% of its customers are looking to buy a sport utility vehicle (SUV), hence p = 0.47.
  • A sample of 61 customers is taken, hence n = 61.

The mean and the standard error are given by:

\mu = p = 0.47

s = \sqrt{\frac{p(1 - p)}{n}} = \sqrt{\frac{0.47(0.53)}{61}} = 0.0639

The probability that less than 40% of the sample are looking to buy an SUV is the <u>p-value of Z when X = 0.4</u>, hence:

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

By the Central Limit Theorem

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

Z = \frac{0.4 - 0.47}{0.0639}

Z = -1.095

Z = -1.095 has a p-value of 0.1368.

0.1368 = 13.68% probability that less than 40% of the sample are looking to buy an SUV.

To learn more about the <em>normal distribution and the central limit theorem</em>, you can check brainly.com/question/24663213

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