Final answer:
The probability of the stock price closing above $203.267 is approximately 0.1587. The probability of the stock price closing below $210.407 is approximately 0.9772. The probability of the stock price closing between $181.84 and $210.407 is approximately 0.9521. A closing price above $208 would be more unusual compared to a closing price below $180.
Step-by-step explanation:
To find the probabilities, we need to standardize the values using the z-score formula:
z = (x - mean) / standard deviation
a) To find the probability of the closing price being above $203.267, we first calculate the z-score:
z = (203.267 - 196.12) / 7.14 = 1.00
Using a standard normal distribution table or calculator, we find that the probability of a z-score being greater than 1.00 is approximately 0.1587. Therefore, the probability is approximately 0.1587.
b) To find the probability of the closing price being below $210.407, we calculate the z-score:
z = (210.407 - 196.12) / 7.14 = 2.00
The probability of a z-score being less than 2.00 is approximately 0.9772. Therefore, the probability is approximately 0.9772.
c) To find the probability of the closing price being between $181.84 and $210.407, we need to calculate the z-scores for both values:
z1 = (181.84 - 196.12) / 7.14 = -1.99
z2 = (210.407 - 196.12) / 7.14 = 2.00
Using the standard normal distribution table or calculator, we find that the probability of a z-score being between -1.99 and 2.00 is approximately 0.9754 - 0.0233 = 0.9521. Therefore, the probability is approximately 0.9521.
d) To determine which is more unusual, a closing price above $208 or below $180, we need to calculate the z-scores:
For $208: z = (208 - 196.12) / 7.14 = 1.66
For $180: z = (180 - 196.12) / 7.14 = -2.26
Based on the z-scores, a closing price above $208 would be more unusual as the corresponding z-score is smaller than the z-score for $180. The z-score of -2.26 indicates a more extreme value.