Fluctuation in the prices of precious metals such as gold have been empirically shown to be well approximated by a normal distribution when observed over short interval of time. In May 1995, the daily price of gold (1 troy ounce) was believed to have a mean of $383 and a standard deviation of $12. A broker, working under these assumptions, wanted to find the probability that the price of gold the next day would be between $394 and $399 per troy ounce. In this eventuality, the broker had an order from a client to sell the gold in the client's portfolio. What is the probability that the client's gold will be sold the next day?
To find the probability that the price of gold will be between $394 and $399 per troy ounce the next day, we can use the normal distribution.
1. Calculate the standard score (z-score) for each price:
- For $394: z = (394 - 383) / 12
- For $399: z = (399 - 383) / 12
2. Look up the corresponding probabilities (area under the normal curve) for these z-scores in a standard normal distribution table or use a calculator. Let's denote the probabilities as P1 and P2.
3. The probability that the price of gold will be between $394 and $399 is given by the difference between P2 and P1: P2 - P1.
By following these steps, we can find the probability that the client's gold will be sold the next day.
Note: When using a normal distribution table, the z-score represents the number of standard deviations from the mean.
To find the probability that the price of gold will be between $394 and $399 per troy ounce the next day, we need to calculate the z-scores for these prices and use the standard normal distribution.
The z-score formula is calculated as follows:
z = (x - μ) / σ
Where:
x is the given value,
μ is the mean, and
σ is the standard deviation.
For $394:
z1 = ($394 - $383) / $12
For $399:
z2 = ($399 - $383) / $12
To find the probability associated with these z-scores, we need to look up the corresponding cumulative probabilities in the standard normal distribution table. Subtracting the value of the smaller z-score from the larger z-score will give us the probability between the two values.
Let's calculate the probabilities step-by-step:
1. Calculate the z-scores:
z1 = ($394 - $383) / $12
z2 = ($399 - $383) / $12
2. Consult the standard normal distribution table and find the cumulative probabilities for these z-scores.
From the standard normal distribution table, we can find that the probability corresponding to z1 is approximately 0.7475, and the probability corresponding to z2 is approximately 0.8413.
3. Calculate the probability between the two z-scores:
P($394 ≤ X ≤ $399) = P(z1 ≤ Z ≤ z2)
= P(z2) - P(z1)
The probabilities can be calculated as follows:
P(z2) = 0.8413
P(z1) = 0.7475
P($394 ≤ X ≤ $399) = P(z2) - P(z1)
= 0.8413 - 0.7475
= 0.0938
Therefore, the probability that the price of gold will be between $394 and $399 per troy ounce the next day is approximately 0.0938, or 9.38%.