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HACTEHA [7]
1 year 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]1 year 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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Step-by-step explanation:

In order to solve this problem we will attribute variables to the numbers, the first one will be "x" and the second one will be "y". From the first sentence we know that the subtraction of the two numbers is equal to 15, so we have:

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