At a certain high school, the distribution of backpack weight is approximately normal with mean 19.7 pounds and standard deviation 3.1 pound

Question

At a certain high school, the distribution of backpack weight is approximately normal with mean 19.7 pounds and standard deviation 3.1 pounds. A random sample of 5 backpacks will be
weight, in pounds, of each backpack will be recorded.
For samples of size 5, which of the following is the best interpretation of Plä > 22)
0.05?
The probability that each of the 5 backpacks selected will have a weight above 22 pounds is approximately 0.05.
The probability that each of the 5 backpacks selected will have a weight above 19.7 pounds is approximately 0.05
The probability that the population mean is greater than 22 pounds is approximately 0.05.
For all samples of size 5, approximately 5% of the sample will have a probability greater than 22 pounds,
For all samples of size 5, the probability that the sample mean will be greater than 22 pounds is approximately 0.05.

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Evelyn 2 weeks 2021-09-28T08:20:03+00:00 1 Answer 0

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    2021-09-28T08:21:39+00:00

    Answer:

    For all samples of size 5, the probability that the sample mean will be greater than 22 pounds is approximately 0.05.

    Step-by-step explanation:

    To solve this question, the normal probability distribution and the central limit theorem were used.

    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.

    For samples of size 5, which of the following is the best interpretation of Plä > 22) = 0.05

    No matter the size of the sample, the mean will be the same.

    However, the standard deviation will be changed.

    P(ä > 22) is the probability of the sample means being above 22.

    So the correct interpretation is:

    For all samples of size 5, the probability that the sample mean will be greater than 22 pounds is approximately 0.05.

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