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According to a PNC Financial Independence Survey released in March 2012, today’s adults in their 20’s “hold an average debt of about $45,000, which includes everything from cars to credit cards to student loans to mortgages. (USA TODAY, April 24, 2012). Suppose that the current distribution of debts of all U.S. adults in their 20’s has a mean of $45,000 and a standard deviation of $12,720. Find the probability that the average debt of a random sample of 144 U.S. adults in their 20’s is: a) Less than $42,600 b) More than $46,240 c) $43,190 to $46,980

Sagot :

Using the normal probability distribution and the central limit theorem, it is found that the probabilities are given by:

a) 0.0119 = 1.19%.

b) 0.121 = 12.1%.

c) 0.9257 = 92.57%.

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, the sampling distribution of sample means of size n has standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].

In this problem:

  • The mean is of [tex]\mu = 45000[/tex].
  • The standard deviation is of [tex]\sigma = 12720[/tex].
  • A sample of 144 is taken, hence [tex]n = 144, s = \frac{12720}{\sqrt{144}} = 1060[/tex].

Item a:

The probability is the p-value of Z when X = 42600, hence:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

By the Central Limit Theorem:

[tex]Z = \frac{X - \mu}{s}[/tex]

[tex]Z = \frac{42600 - 45000}{1060}[/tex]

[tex]Z = -2.26[/tex]

[tex]Z = -2.26[/tex] has a p-value of 0.0119.

The probability is of 0.0119 = 1.19%.

Item b:

The probability is the 1 subtracted by the p-value of Z when X = 46240, hence:

[tex]Z = \frac{X - \mu}{s}[/tex]

[tex]Z = \frac{46240 - 45000}{1060}[/tex]

[tex]Z = 1.17[/tex]

[tex]Z = 1.17[/tex] has a p-value of 0.879.

1 - 0.879 = 0.121.

The probability is of 0.121 = 12.1%.

Item c:

The probability is the p-value of Z when X = 46980 subtracted by the p-value of Z when X = 43190, hence:

X = 46980:

[tex]Z = \frac{X - \mu}{s}[/tex]

[tex]Z = \frac{46980 - 45000}{1060}[/tex]

[tex]Z = 1.87[/tex]

[tex]Z = 1.87[/tex] has a p-value of 0.9693.

X = 43190:

[tex]Z = \frac{X - \mu}{s}[/tex]

[tex]Z = \frac{43190 - 45000}{1060}[/tex]

[tex]Z = -1.71[/tex]

[tex]Z = -1.71[/tex] has a p-value of 0.0436.

0.9693 - 0.0436 = 0.9257.

The probability is of 0.9257 = 92.57%.

To learn more about the normal probability distribution and the central limit theorem, you can check https://brainly.com/question/24663213

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