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Let π be a normal random variable.
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Find the probability that π is greater than π plus 0.71π.
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Remember the graph of a curve representing the normal distribution with a mean of π and a standard deviation of π is bell-shaped and symmetric about the mean.
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And the total area under the curve is 100 percent or one.
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A sketch of the curve can be a really useful way to help decide how to answer a problem about normally distributed data.
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In this case, weβre looking to find the probability that π is greater than π plus 0.71π; thatβs this shaded region.
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We know it must sit above the mean in our bell curve because the standard deviation canβt be negative.
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Once weβve established this, the next step with most normal distribution questions is to calculate the π-value.
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This is a way of scaling our data or standardising it in what becomes a standard normal distribution.
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Once we complete this step, we can work from a single standard normal table.
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Now, it doesnβt really matter that we havenβt got a numerical value for the mean π or the standard deviation π of this data set.
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Letβs see what happens when we substitute everything we know into our formula for the π-value.
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Our value for π is π plus 0.71π and then we subtract π and we divide through by π.
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π minus π is zero.
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So our formula simplifies somewhat to 0.71π all divided by π.
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However, we can simplify a little further by dividing through by π.
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And we get π is equal to 0.71.
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So weβre looking to find the probability that π is greater than 0.71 since in the original question, it was asking us to find the probability that π is greater than π plus 0.71π.
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Our standard normal table though only gives probabilities between zero and π.
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In this case, thatβs this side of the curve.
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So we find the probability that π is greater than 0.71 by subtracting the probability that itβs less than 0.71 from one because we said that the area under the curve is 100 percent or one whole.
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Looking up a π-value of 0.71 in our standard normal table and we can see that the probability that π is less than 0.711 is 0.7611.
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That means the probability that π is greater than 0.71 is one minus 0.7611; thatβs 0.2389.
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That means the probability that π is greater than π plus 0.71π is equal to 0.2389.