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Statistics

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Apr 3, 2024

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Use the following information to answer the next three questions. An unknown distribution has the following parameters: μx =45 and σx = 8. A sample size of 50 is drawn randomly from the population. Find P ( ΣX >2,400). Find Σ X where z=-2. Find the 80th percentile for the sum of the 50 values of x. 1. Find P ( ΣX >2,400) To solve these questions, we need to use the properties of the normal distribution. Given that we have the population parameters, we can calculate the mean and standard deviation of the sample distribution as follows: First, the mean and standard deviation of the sum of the 50 values. The mean of the sum (μΣX) is equal to n * μx, where n is the sample size and μx is the mean of the population. The standard deviation of the sum (σΣX) is equal to sqrt(n) * σx, where n is the sample size and σx is the standard deviation of the population(Hargrave, 2023). So, μΣX = n * μx = 50 * 45 = 2250 σΣX = sqrt(n) * σx = sqrt(50) * 8 = 35.36 Now, we can find the z-score for ΣX = 2400 using the formula: z = (ΣX - μΣX) / σΣX = (2400 - 2250) / 35.36 = 4.24 It can be approximated with a value of 0 by looking at the area to the right of z = 4.24 on the standard normal distribution. 2. Find ΣX where z=-2 We can use the z-score formula to find ΣX: ΣX = z * σΣX + μΣX = -2 * 35.36 + 2250 = 2179.28 3. Find the 80th percentile for the sum of the 50 values of x The z-score for the 80th percentile is approximately 0.84. We can use the z-score formula to find the 80th percentile: ΣX = z * σΣX + μΣX = 0.84 * 35.36 + 2250 = 2299.70 the 80th percentile for the sum of the 50 values of x is approximately 2299.70(Hargrave, 2023). Reference

Hargrave, M. (2023, December 13). Standard Deviation Formula and Uses vs. Variance . Investopedia. https://www.investopedia.com/terms/s/standarddeviation.asp

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