Suppose that a random sample of size 36 is to be selected from a population with mean 43 and standard deviation 6. What is the approximate p

Question

Suppose that a random sample of size 36 is to be selected from a population with mean 43 and standard deviation 6. What is the approximate probability that X will be more than 0.5 away from the population mean?

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Josie 2 months 2021-10-17T15:13:38+00:00 1 Answer 0 views 0

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    2021-10-17T15:14:56+00:00

    Answer:

    61.70% approximate probability that X will be more than 0.5 away from the population mean

    Step-by-step explanation:

    To solve this question, we have 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 random variable X, with mean \mu and standard deviation \sigma, a large sample size can be approximated to a normal distribution with mean \mu and standard deviation s = \frac{\sigma}{\sqrt{n}}

    In this problem, we have that:

    \mu = 36, \sigma = 6, n = 36, s = \frac{6}{\sqrt{36}} = 1

    What is the approximate probability that X will be more than 0.5 away from the population mean?

    This is the probability that X is lower than 36-0.5 = 35.5 or higher than 36 + 0.5 = 36.5.

    Lower than 35.5

    Pvalue of Z when X = 35.5. So

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

    By the Central Limit Theorem

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

    Z = \frac{35.5 - 36}{1}

    Z = -0.5

    Z = -0.5 has a pvalue of 0.3085.

    30.85% probability that X is lower than 35.5.

    Higher than 36.5

    1 subtracted by the pvalue of Z when X = 36.5. SO

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

    Z = \frac{36.5 - 36}{1}

    Z = 0.5

    Z = 0.5 has a pvalue of 0.6915.

    1 – 0.6915 = 0.3085

    30.85% probability that X is higher than 36.5

    Lower than 35.5 or higher than 36.5

    2*30.85 = 61.70

    61.70% approximate probability that X will be more than 0.5 away from the population mean

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