Final answer:
The probabilities for the normally distributed profit X of a store with a mean of $360 and a standard deviation of $50 can be found using Z scores and standard normal distribution tables. The probability of X being greater than $400, less than $300, equal to $420, and less than $360 can be determined as described, with P(X = $420) being effectively zero due to the continuous nature of the normal distribution.
Step-by-step explanation:
The random variable X represents the profit made on a randomly selected day by a certain store, which is normally distributed with a mean of $360 and a standard deviation of $50. To find the probabilities asked, we can use a standard normal distribution table or a calculator after converting the X values to Z scores, which are standardized values that represent the number of standard deviations away from the mean.
- a) To find P(X > $400), we first calculate the Z score for $400: Z = ($400 - $360) / $50 = 0.8. Using a Z table, we find the probability corresponding to Z = 0.8 and subtract it from 1 to get P(X > $400).
- b) To find P(X < $300), we calculate the Z score for $300: Z = ($300 - $360) / $50 = -1.2. We use the Z table to find the probability for Z = -1.2, which gives us P(X < $300).
- c) Since the normal distribution is continuous, the probability that X equals any single value, such as P(X = $420), is effectively zero.
- d) For P(X < $360), which is exactly the mean, this probability is always 0.5 because the normal distribution is symmetric about the mean.
Note that for the calculations to be accurate, the normal distribution table, software, or a calculator is required.