The mean of a normal probability distribution is 500 and the standard deviation is 10. About 95 percent of the observations lie between what two values?
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A. B. C. D. E.C
95% lie between plus and minus two standard deviations of the mean.
To determine the values between which 95 percent of the observations lie in a normal distribution, we need to use the concept of z-scores.
The z-score represents the number of standard deviations an observation is away from the mean. In a normal distribution, approximately 95 percent of the observations lie within two standard deviations of the mean.
Given that the mean of the distribution is 500 and the standard deviation is 10, we can calculate the z-scores for the values that correspond to two standard deviations away from the mean.
To find the z-scores, we use the formula: z = (X - μ) / σ
Where: z = z-score X = observed value μ = mean σ = standard deviation
For the lower bound, we calculate the z-score using X = μ - 2σ: z_lower = (X - μ) / σ = (475 - 500) / 10 = -2.5
For the upper bound, we calculate the z-score using X = μ + 2σ: z_upper = (X - μ) / σ = (525 - 500) / 10 = 2.5
Now, we need to find the corresponding values for these z-scores.
Using a standard normal distribution table or a calculator, we can find that the area under the curve to the left of a z-score of -2.5 is approximately 0.0062. This means that approximately 0.62 percent of the observations lie below the lower bound. To find the value corresponding to this z-score, we can multiply it by the standard deviation and add it to the mean: Lower bound = μ + (z_lower * σ) = 500 + (-2.5 * 10) = 475
Similarly, the area under the curve to the left of a z-score of 2.5 is also approximately 0.9938. This means that approximately 99.38 percent of the observations lie below the upper bound. To find the value corresponding to this z-score, we can multiply it by the standard deviation and add it to the mean: Upper bound = μ + (z_upper * σ) = 500 + (2.5 * 10) = 525
Therefore, we can conclude that about 95 percent of the observations lie between 475 and 525, which corresponds to answer choice B: 475 and 525.