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1 April, 00:29

A large school district in southern California asked all of its eighth-graders to measure the length of their right foot at the beginning of the school year, as part of a science project. The data show that foot length is approximately Normally distributed, with a mean of 23.4 cm and a standard deviation of 1.7 cm. Suppose that 25 eighth-graders from this population are randomly selected. Approximately what is probability that the sample mean foot length is less than 23 cm?

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  1. 1 April, 02:45
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    The probability of the sample mean foot length less than 23 cm is 0.120

    Step-by-step explanation:

    * Lets explain the information in the problem

    - The eighth-graders asked to measure the length of their right foot at

    the beginning of the school year, as part of a science project

    - The foot length is approximately Normally distributed, with a mean of

    23.4 cm

    ∴ μ = 23.4 cm

    - The standard deviation of 1.7

    ∴ σ = 1.7 cm

    - 25 eighth-graders from this population are randomly selected

    ∴ n = 25

    - To find the probability of the sample mean foot length less than 23

    ∴ The sample mean x = 23, find the standard deviation σx

    - The rule to find σx is σx = σ/√n

    ∵ σ = 1.7 and n = 25

    ∴ σx = 1.7/√25 = 1.7/5 = 0.34

    - Now lets find the z-score using the rule z-score = (x - μ) / σx

    ∵ x = 23, μ = 23.4, σx = 0.34

    ∴ z-score = (23 - 23.4) / 0.34 = - 1.17647 ≅ - 1.18

    - Use the table of the normal distribution to find P (x < 23)

    - We will search in the raw of - 1.1 and look to the column of 0.08

    ∴ P (X < 23) = 0.119 ≅ 0.120

    * The probability of the sample mean foot length less than 23 cm is 0.120
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