# Video: MATH-STATS-2018-S1-Q05

If π is a standard normal random variable such that π(βπ β€ π β€ π) = 0.874, find π.

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### Video Transcript

If π is a standard normal random variable such that the probability that negative π is less than or equal to π which is less than or equal to π is 0.874, find π.

If π is a standard normal random variable, as weβre told in the question, then this means that it has a normal distribution with a mean of zero and a standard deviation of one. The standard normal distribution is a bell-shaped curve symmetrical about its mean. So, in this case, thatβs zero. The area under the full curve is one. And the area to the left of a particular value gives the probability of a value from this distribution being less than or equal to that value.

In this question, weβre told that the probability that π is between negative π and π is 0.874. So thatβs this area that Iβve shaded in orange. We need to convert this to an area to the left of a particular value so that we can use our standard normal tables to find the value of π. Remember that this distribution is symmetrical, which means that the two orange areas, either side of the mean, are equal. So theyβre each equal to half of 0.874 which is 0.437. The whole area to the left of the mean is 0.5. Thatβs half of the total area of one, again due to the symmetry of the distribution. This means that the area to the left of the value π is 0.5, thatβs for the pink region, plus 0.437, for the orange region, giving a total of 0.937.

Remember that the area to the left of a particular value represents the probability that π, our standard normal random variable, is less than or equal to this value. So we now know that the probability that π is less than or equal to π is 0.937. We can use our standard normal tables to look up this probability and find the π-score, which will be the value of π, associated with it.

Now, here is an extract from those standard normal tables. And we can see that a probability of 0.937, or 0.9370, is located here. Moving horizontally across from this probability, we can see that the π-score is 1.50. And then moving upwards from this probability, we can see that thereβs a three in the second decimal place, the hundredths column. So we add 0.03 to 1.50. And this tells us that the value of π is 1.53.