Directions: Select one of the following answer choices.Q. A random variable Y is normally distributed with a mean of 200 and a standard deviation of 10.Quantity A: The probability of the event that the value of Y is greater than 220Quantity B: 1/6a)Quantity A is greater.b)Quantity B is greater.c)The two quantities are equal.d)The relationship cannot be determined from the information given.Correct answer is option 'B'. Can you explain this answer?

Question

Answer

To determine whether Quantity A or Quantity B is greater, we need to compare the probability of the event that the value of Y is greater than 220 (Quantity A) with 1/6 (Quantity B).

Given information:
- Mean of Y = 200
- Standard deviation of Y = 10

We know that Y follows a normal distribution. In a normal distribution, we can use the z-score formula to calculate the probability of an event.

The z-score formula is given by:
z = (X - μ) / σ

where X is the value we are interested in, μ is the mean, and σ is the standard deviation.

Let's calculate the z-score for the event that Y is greater than 220.

z = (220 - 200) / 10
z = 20 / 10
z = 2

To find the probability corresponding to a z-score of 2, we can refer to the standard normal distribution table or use a calculator. In this case, the probability is approximately 0.9772.

Now, let's compare Quantity A and Quantity B:

Quantity A: The probability of the event that the value of Y is greater than 220 is 0.9772.

Quantity B: 1/6

Since 0.9772 is greater than 1/6, we can conclude that Quantity A is greater than Quantity B.

Therefore, the correct answer is option 'b' - Quantity B is greater.

Answer

This problem involves a normal distribution with mean 200 and standard deviation 10. Thus, the value of 210 is 1 standard deviation above the mean, and the value of 220 is 2 standard deviations above the mean. To compare Quantity A with Quantity B, it is not necessary to exactly determine the probability of the event that the value of Y is greater than 220. Remember that in any normal distribution, almost all of the data values, or about 95% of them, fall within 2 standard deviations on either side of the mean. This means that less than 5% of the values in this distribution will be greater than 220. Thus, the probability of the event that the value of Y is greater than 220 must be less than 5%, or 1/20, and this is certainly less than 1/6. The correct answer is Choice B.

Another approach to this problem is to draw a normal curve, or “bell-shaped curve,” that represents the probability distribution of the random variable Y, as shown in the figure below.

The curve is symmetric about the mean 200. The values of 210, 220, and 230 are equally spaced to the right of 200 and represent 1, 2, and 3 standard deviations, respectively, above the mean. Similarly, the values of 190, 180, and 170 are 1, 2, and 3 standard deviations, respectively, below the mean. Quantity A, the probability of the event that the value of Y is greater than 220, is equal to the area of the shaded region as a fraction of the total area under the curve.

From the figure, you can see that the area under the normal curve has been divided into 6 regions and that these regions are not equal in area. The shaded region is one of the two smallest of the 6 regions, so its area must be less than 1/6 of the total area under the curve. The correct answer is Choice B.

Similar GRE Doubts

Directions: Select one of the following answer choices.Q. p is the probability that event E will occur, and s is the probability that event E will not occur.Quantity A: p + sQuantity B: ps

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