a) Fact: the region under the standard normal curve that lies to the left of -1.

Question

a) Fact: the region under the standard normal curve that lies to the left of -1.28 has area 0.100273. Without consulting a table or a calculator giving areas under the standard normal curve, determine the area under the standard normal curve that lies to the right of 1.28 b) Which property of the standard normal curve allowed you to answer part a)? A. The standard normal curve extends indefinitely in both directions B. Almost all the area under the standard normal curve lies between -3 and 3 C. The standard normal curve is symmetric about 0 D. The total area under the curve is 1 E. None of the above

Explanation / Answer

Solution

Property of Normal Distribution.

One important property of Normal Distribution is that it is symmetric about its mean. i.e.,

If X ~ N(µ, 2), where µ = mean and 2 = variance, P(X < µ - d) = P(X > µ + d)……..(1)

or put in a different way, P(µ - d < X < µ) = P(µ < X < µ + d)………………………..(2)

We are given, that the area under Standard Normal Curve to the left of – 1.28 = 0.100273.

Now, area under the curve represents probability and hence the given information is:

P(Z < - 1.28) = 0.100273 where Z ~ N(0, 1).

Substituting µ = 0 and d = 1.28 in (1), we have P(Z < 0 – 1.28) = P(Z > 0 + 1.28) or

P(Z < - 1.28) = P(Z > 1.28)………………………………………………………..(3)

Since area under the curve represents probability, (3) =>

Area to the left of – 1.28 = area to the right of 1.28.

Thus, the answer is 0.100273 ANSWER 1    

As already mentioned above, the property used to get this result is ‘Standard Normal Curve is symmetric about mean zero’ Option (D) ANSWER 2

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