Let’s define X as the astronomical error, which is normally distributed with mean 0 km and standard deviation 1,000 km. Now, you have to find the probability that -500 < X < 500, i.e. P(-500 < X < 500).
Question
Let’s define X as the astronomical error, which is normally distributed with mean 0 km and standard deviation 1,000 km. Now, you have to find the probability that -500 < X < 500, i.e. P(-500 < X < 500).
Solution
To solve this problem, we need to use the properties of the standard normal distribution.
Step 1: Standardize the random variable X. We do this by subtracting the mean and dividing by the standard deviation. This gives us Z = (X - μ) / σ. In this case, μ = 0 and σ = 1000, so Z = X / 1000.
Step 2: Substitute the values of X into the inequality -500 < X < 500 to get -0.5 < Z < 0.5.
Step 3: Look up the values of -0.5 and 0.5 in the standard normal distribution table. The value for 0.5 is 0.6915 and the value for -0.5 is 0.3085.
Step 4: Subtract the value of -0.5 from the value of 0.5 to get the probability that -500 < X < 500. This gives us 0.6915 - 0.3085 = 0.3830.
So, P(-500 < X < 500) = 0.3830, or 38.30%.
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Hence, what is the probability that the astronomer under- or over-estimates the distance by less than 500 km?
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