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If π§ is a standard normally distributed variable where the probability that negative π is less than or equal to π§ is less than or equal to π equals 0.8664, find π.
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Weβre told first of all that π§ has a normal distribution.
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And a normal distribution is defined by its two parameters, its mean π and its standard deviation πΏ.
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Weβre further told that π§ has the standard normal distribution which means that its mean π is equal to zero and its standard deviation πΏ is equal to one.
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So we can write that π§ follows a normal distribution zero, one squared.
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Weβre also told the probability that π§ lies between two values, negative π and π.
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And we want to work out the value of π.
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To do so, letβs recall how we find probabilities for a normal distribution.
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We recall first of all that the normal distribution is a bell-shaped curve symmetrical about its mean π.
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The area below the full curve is equal to one.
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And the area to the left of a particular value gives the probability that the random variable π§ takes a value less than or equal to this value.
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We can work out these probabilities by first calculating as π§-score.
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That tells us how many standard deviations a particular value is away from the mean.
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If we were working with a normal distribution other than the standard normal, we would calculate the π§-score for any particular value π₯ using the formula π§ equals π₯ minus π over πΏ.
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We subtract the mean from the value π₯ and then divide by the standard deviation.
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However for the standard normal, the mean π is zero.
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And the standard deviation πΏ is one.
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So we have π₯ minus zero over one which is just π₯.
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For a standard normal, the π§-score of a particular value is just that value itself.
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Once weβve calculated the π§-score for a particular value, we can use our standard normal tables in order to look at the probability associated with that π§-score.
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However, we havenβt actually been given the probability that π§ is less than or equal to a particular value.
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Instead, weβve been given the probability that π§ is between two values, negative π and π.
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So how can we work backwards from knowing this probability to calculating the value of π?
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Well, there are actually a couple of different methods that we can use depending on the type of statistical tables that are available to us.
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For the first method, weβll use tables which give the probability that the standard normally distributed variable π§ is between zero and a value π§.
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That would correspond to just the area now shaded in pink on our normal distribution curve.
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So we need to know the probability associated with just this area.
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Well, to work this out, we recall again that our normal distribution is symmetrical about its mean, which means that the orange and pink areas are exactly the same.
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Theyβre each half of the probability of 0.8664.
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So theyβre both equal to 0.4332.
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Now, here is that first type of statistical table.
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And we need to look up our probability of 0.4332.
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That probability is located here in our table.
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And if we look horizontally across, we see that it is associated with the π§-score of 1.5.
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If we look vertically upwards, then we can see that the contribution in the second decimal place is zero.
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That tells us that the π§-score which, in this case, is just a value of π is 1.50 or 1.5.
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So we have our answer π equals 1.5.
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This tells us that the probability that π§ is between 1.5 standard deviations below the mean and 1.5 standard deviations above the mean is 0.8664.
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But there is another method that we can use.
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This is to use tables which give the probability that π§ is less than or equal to a particular value.
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On our normal distribution curve, this would mean weβre looking at all of the area to the left of that value π.
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So thatβs the orange area and the pink area combined.
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We know that the pink area corresponds to a probability 0.4332.
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But what about our new orange area?
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Well, we recall again that the normal distribution is symmetrical about its mean which means that the area on either side of the mean is half of the total area of one.
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The orange area is, therefore, equal to 0.5.
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This means that the total area to the left of π and, therefore, the probability that π§ is less than or equal to π is 0.5 plus 0.4332 which is 0.9332.
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If we were to look up a probability of 0.9332 in our second type of statistical tables, we would again see that this is associated with the π§-score of 1.5.
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We found then that if π§ is a standard normally distributed variable where the probability that negative π is less than or equal to π§ is less than or equal to π is 0.8664, then π is equal to 1.5.