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π is a normal random variable whose mean is zero and standard deviation is π.
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If the probability that π is less than or equal to ππ is 0.877, find the value of π.
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Weβve been told then that π is a normal random variable with mean zero and standard deviation π.
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We can express this then using the usual notation for normal distribution.
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π has a normal distribution with mean zero and standard deviation π.
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It, therefore, has a variance of π squared.
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Now, weβre told that the probability that π is less than or equal to some value, ππ, is 0.877.
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And we want to work out the value of π.
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First, we recall that a normal distribution is a bell-shaped curve symmetrical about its mean, π, which in this question is zero.
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The area below the full curve is one.
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And the area to the left of any particular value gives the probability that π is less than or equal to that value.
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In our case, the value is ππ, and the probability is 0.877.
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Now, in order to work out these probabilities for a normal distribution, we use a π§-score, which is found by subtracting the mean π from that particular value π and then dividing by the standard deviation π.
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This tells us how many standard deviations a particular value π is from the mean π.
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So to work out the π§-score for the value of ππ in this distribution, we subtract the mean zero and then divide by the standard deviation π, giving ππ minus zero over π.
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ππ minus zero is just ππ, and then dividing by π gives π.
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So the π§-score associated with a value of ππ is just π.
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This make sense.
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Remember, a π§-score tells us the number of standard deviations that a value is away from the mean.
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And if the mean is zero, then a value of ππ will be π lots of π, or π standard deviations, above the mean.
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We would then use our standard normal distribution tables to work out the probability associated with a particular π§-score.
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But, in this question, weβre going the other way.
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We know the probability 0.877.
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And we want to work backwards to find the corresponding π§-score, which gives the value of π.
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Our probability of 0.877 or 0.8770 is located here in the table.
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Looking across, we see that this is associated with a π§-score of 1.10.
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And then looking upwards, we see that there is an additional 0.06.
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So we need to include a six in the second decimal place.
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So we find then that the π§-score associated with a probability of 0.8770, and therefore the value of π, is 1.16.
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This means that for a normal random variable with a mean zero, the probability of that random variable π taking a value less than or equal to 1.16 standard deviations above the mean is 0.877.