Question

Suppose that the value of a stock varies each day from $10.82 to $33.17 with a uniform distribution. Find the upper quartile; 25% of all days the stock is above what value?

Answer 1

To find the upper quartile, we need to first find the median of the stock prices, which is the value that divides the distribution into two equal parts. The midpoint of the distribution is:

Midpoint = (10.82 + 33.17) / 2 = 22.995

Now, we can find the upper quartile, which is the median of the upper half of the distribution. The upper half of the distribution ranges from the midpoint to the highest value of 33.17. Therefore, we calculate the median of this range as follows:

Upper quartile = (22.995 + 33.17) / 2 = 28.0825

So, the upper quartile of the stock prices is $28.08.

To find the value above which the stock is priced 25% of the time, we need to find the 75th percentile of the distribution. Since the distribution is uniform, we can use the formula for the percentile as follows:

Percentile rank = (percentile / 100) = (value - minimum) / (maximum - minimum)

Solving for the value, we get:

value = minimum + percentile rank x (maximum - minimum)

For the 75th percentile, we have:

value = 10.82 + 0.75 x (33.17 - 10.82) = 28.49

Therefore, the stock is priced above $28.49 on 25% of all days.

Answer 2

The range of the stock prices is from $10.82 to $33.17. To find the upper quartile, we need to find the value of the stock price that is greater than 75% of the data.

The distance between the minimum value and the maximum value is:

$33.17 - $10.82 = $22.35

To find the upper quartile, we need to find the value that is three-quarters of the distance above the minimum value:

$10.82 + 0.75($22.35) = $26.69

Therefore, the upper quartile of the stock price is $26.69.

To find the value above which the stock price is higher 25% of the time, we need to find the value that is 75% of the distance above the minimum value:

$10.82 + 0.25($22.35) = $15.62

Therefore, 25% of the time, the stock price is above $15.62.