Final answer:
To find the probability that the mean of the 40 applicants is above 215, we use the Central Limit Theorem to calculate the standard deviation of the sampling distribution. Then, we convert the desired value into a standard score and find the probability using a z-table or calculator.
Step-by-step explanation:
To solve this problem, we need to use the Central Limit Theorem. According to the Central Limit Theorem, the sampling distribution of sample means will be approximately normally distributed, with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size.
In this case, the mean of the population is 200 and the standard deviation is 50. The sample size is 40. So, the mean of the sampling distribution of sample means will still be 200, but the standard deviation will be 50 divided by the square root of 40. Therefore, the standard deviation of the sampling distribution is approximately 7.07.
To find the probability that the mean of the 40 applicants is above 215, we can convert this into a standard score using the formula z = (x - μ) / σ, where x is the value we are interested in, μ is the mean of the sampling distribution, and σ is the standard deviation of the sampling distribution. Plugging in the values, we get z = (215 - 200) / 7.07 = 2.12.
Next, we need to find the probability of getting a z-score above 2.12. We can use a z-table or a calculator to find the probability. Looking up the z-score in a standard normal distribution table, the probability of getting a z-score above 2.12 is approximately 0.0187.