A population has mean 187 and standard deviation 32. If a random sample of 64 observations is selected at random from this population, what

Question

A population has mean 187 and standard deviation 32. If a random sample of 64 observations is selected at random from this population, what is the probability that the sample average will be less than 182

in progress 0
Valentina 1 month 2021-10-18T06:06:55+00:00 1 Answer 0 views 0

Answers ( )

    0
    2021-10-18T06:08:32+00:00

    Answer:

    11.51% probability that the sample average will be less than 182

    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 = 187, \sigma = 32, n = 64, s = \frac{32}{\sqrt{64}} = 4

    What is the probability that the sample average will be less than 182

    This is the pvalue of Z when X = 182. So

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

    By the Central Limit Theorem

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

    Z = \frac{182 - 187}{4}

    Z = -1.2

    Z = -1.2 has a pvalue of 0.1151

    11.51% probability that the sample average will be less than 182

Leave an answer

45:7+7-4:2-5:5*4+35:2 =? ( )