The melting point of gold is known to be approximately 1,060 degrees centigrade. This is, of course, an average figure, for unavoidable ‘experimental error’ causes more or less variation from this figure whenever the test is actually performed. The best measure of these variations is the standard deviation,. Suppose this has been calculated from a large series of tests, and found to be 3 degrees centigrade. Due to the recent increase in the price of gold, special attention may be necessary to confirm that a product is actually gold. Suppose an investigator is analyzing an unknown metal, and a test shows its melting point to be 1,066 degrees centigrade.
a)Calculate the zscore for this observation of 1,066.
b) What is the probability that a sample of gold would show a melting point as different from its average as 1,066 degrees centigrade?
c) Is it likely that this unknown metal is gold?
a) To calculate the z-score for an observation, you need to use the formula:
z = (x - μ) / σ
where:
- x is the observation (1,066 degrees centigrade in this case)
- μ is the mean or average (1,060 degrees centigrade)
- σ is the standard deviation (3 degrees centigrade)
Plug in the values into the formula:
z = (1,066 - 1,060) / 3 = 2 / 3 ≈ 0.67
So, the z-score for this observation is approximately 0.67.
b) To find the probability of an observation deviating from the average, you can use the z-table or a statistical software.
From the z-table, find the area under the normal distribution curve corresponding to the z-score of 0.67. The z-table provides the corresponding probability, which is 0.7486.
However, since we want to find the probability of a sample of gold showing a melting point as different from its average (not specifically less or greater), we need to consider the probability in both tails of the distribution. Thus, double the probability obtained from the z-table:
P = 2 * 0.7486 = 1.4972
The probability that a sample of gold would show a melting point as different from its average as 1,066 degrees centigrade is approximately 1.4972 or 149.72%.
c) To determine whether the unknown metal is likely to be gold based on the melting point, we need to consider how likely it is for a sample of gold to have a melting point as different as 1,066 degrees centigrade.
The probability we calculated in part b (1.4972 or 149.72%) indicates that it is quite common for a sample of gold to have a melting point that deviates from the average. Therefore, it is not conclusive evidence that the unknown metal is gold based solely on the melting point.
Other tests and analyses would need to be carried out to confirm its identity as gold.