To find the probability that between 55% and 60% of a sample of 150 gold dealers believe that it will be a good year to speculate in South African gold coins, you can use the normal distribution and the given information that 58% of all gold dealers believe that it will be a good year to speculate.
The normal distribution is a continuous probability distribution that is symmetrical around the mean. The mean in this case is 58%, and the standard deviation can be calculated using the formula:
standard deviation = sqrt( (p * (1 - p)) / n )
where p is the proportion of all gold dealers who believe that it will be a good year to speculate (58% in this case) and n is the sample size (150 in this case).
Plugging in the given values, we get:
standard deviation = sqrt( (0.58 * (1 - 0.58)) / 150 ) = 0.042
To find the probability that between 55% and 60% of the sample believe that it will be a good year to speculate, you can use a z-score table to find the z-score for 55% and 60%, and then use these z-scores to find the probability.
The z-score for 55% is (55 - 58) / 0.042 = -7.14, and the z-score for 60% is (60 - 58) / 0.042 = 2.38.
Using a z-score table, you can find that the probability of a z-score between -7.14 and 2.38 is approximately 0.45.
Therefore, the probability that between 55% and 60% of a sample of 150 gold dealers believe that it will be a good year to speculate in South African gold coins is approximately 0.45.