Suppose that the walking step lengths of adult males are normally distributed with a mean of 2.5 feet and a standard deviation of 0.2 feet.

Question

Suppose that the walking step lengths of adult males are normally distributed with a mean of 2.5 feet and a standard deviation of 0.2 feet. A sample of 41 men’s step
lengths is taken
Step 2 of 2: Find the probability that the mean of the sample taken is less than 2.2 feet. Round your answer to 4 decimal places, if necessary,

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Thành Đạt 4 years 2021-08-17T07:47:56+00:00 1 Answers 14 views 0

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  1. Jezebel
    0
    2021-08-17T07:49:52+00:00
    Reply

    Answer:

    0% probability that the mean of the sample taken is less than 2.2 feet.

    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 normal distributions can be solved using the z-score formula.

    In a set with mean \mu and standard deviation \sigma, the z-score 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 p-value, we get the probability that the value of the measure is greater than X.

    Central Limit Theorem

    The Central Limit Theorem establishes 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.

    Mean of 2.5 feet and a standard deviation of 0.2 feet.

    This means that \mu = 2.5, \sigma = 0.2

    Sample of 41

    This means that n = 41, s = \frac{0.2}{\sqrt{41}}

    Find the probability that the mean of the sample taken is less than 2.2 feet.

    This is the p-value of Z when X = 2.2 So

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

    By the Central Limit Theorem

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

    Z = \frac{2.2 - 2.5}{\frac{0.2}{\sqrt{41}}}

    Z = -9.6

    Z = -9.6 has a p-value of 0.

    0% probability that the mean of the sample taken is less than 2.2 feet.

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