The mean rent of a 3-bedroom apartment in Orlando is $1300. You randomly select 10 apartments around town. The rents are normally distribute

Question

The mean rent of a 3-bedroom apartment in Orlando is $1300. You randomly select 10 apartments around town. The rents are normally distributed with a standard deviation of $350. What is the probability that the mean rent is more than $1100?

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3 weeks 2021-12-27T17:30:19+00:00 1 Answer 0 views 0

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    2021-12-27T17:31:57+00:00

    Answer:

    96.49% probability that the mean rent is more than $1100

    Step-by-step explanation:

    To solve this question, we need to understand the normal probability distribution and the central limit theorem.

    Normal probability distribution

    Problems of normally distributed samples are solved using the z-score formula.

    In a set with mean \mu and standard deviation \sigma, the zscore of a measure X is given by:

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

    The Z-score 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. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

    Central Limit Theorem

    The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean \mu and standard deviation \sigma, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean \mu and standard deviation s = \frac{\sigma}{\sqrt{n}}.

    For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

    In this problem, we have that:

    \mu = 1300, \sigma = 350, n = 10, s = \frac{350}{\sqrt{10}} = 110.68

    What is the probability that the mean rent is more than $1100?

    This is 1 subtracted by the pvalue of Z when X = 1100. So

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

    By the Central Limit Theorem

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

    Z = \frac{1100 - 1300}{110.68}

    Z = -1.81

    Z = -1.81 has a pvalue of 0.0351

    1 – 0.0351 = 0.9649

    96.49% probability that the mean rent is more than $1100

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