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Sagot :
Using the normal distribution and the central limit theorem, it is found that the probability is of 0.1368 = 13.68%.
Normal Probability Distribution
In a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
- 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 [tex]\mu = p[/tex] and standard deviation [tex]s = \sqrt{\frac{p(1 - p)}{n}}[/tex], as long as [tex]np \geq 10[/tex] and [tex]n(1 - p) \geq 10[/tex].
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:
[tex]\mu = p = 0.47[/tex]
[tex]s = \sqrt{\frac{p(1 - p)}{n}} = \sqrt{\frac{0.47(0.53)}{61}} = 0.0639[/tex]
The probability that less than 40% of the sample are looking to buy an SUV is the p-value of Z when X = 0.4, hence:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{0.4 - 0.47}{0.0639}[/tex]
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 normal distribution and the central limit theorem, you can check https://brainly.com/question/24663213
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