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26 September, 06:02

A large tank of fish from a hatchery is being delivered to a lake. The hatchery claims that the mean length of fish in the tank is 15 inches, and the standard deviation is 4 inches. A random sample of 31 fish is taken from the tank. Let x be the mean sample length of these fish. What is the probability that x is within 0.5 inch of the claimed population mean? (Round your answer to four decimal places.)

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  1. 26 September, 06:14
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    Step-by-step explanation:

    Let us assume that the length of fishes in the tank is normally distributed. Thus, x is the random variable representing the length of fishes in the tank. Since the population mean and population standard deviation are known, we would apply the formula,

    z = (x - µ) / (σ/√n)

    Where

    x = sample mean

    µ = population mean

    σ = standard deviation

    number of samples

    From the information given,

    µ = 15

    σ = 4

    n = 31

    We want to find the probability that x is between (15 - 0.5) inches and (15 + 0.5) inches. The probability is expressed as

    P (14.5 ≤ x ≤ 15.5)

    For x = 14.5

    z = (14.5 - 15) / (4/√31) = - 0.7

    Looking at the normal distribution table, the probability corresponding to the z score is 0.2420

    For x = 15.5

    z = (15.5 - 15) / (4/√31) = 0.7

    Looking at the normal distribution table, the probability corresponding to the z score is 0.7580

    Therefore,

    P (14.5 ≤ x ≤ 15.5) = 0.7580 - 0.2420 = 0.5160
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