Final answer:
To find the probabilities, calculate the standard error and use it to find the z-score. Then, use a standard normal distribution table or calculator to find the probabilities. As the sample size increases, the probability of the mean exceeding 105 decreases.
Step-by-step explanation:
a. Finding the probability that a randomly selected individual has an IQ greater than 105
To find this probability, we need to calculate the area under the normal distribution curve to the right of 105. We can use a standard normal distribution table or a calculator to find this area. Using a standard normal distribution table, we find that the probability is approximately 0.1587.
b. Finding the probability that a randomly selected sample of size n=25 has a mean greater than 105
To find this probability, we need to calculate the standard error of the sample mean and use it to find the z-score. The z-score can then be used to find the probability using a standard normal distribution table or a calculator. The standard error is calculated by dividing the standard deviation by the square root of the sample size. With a sample size of 25, the standard error is (15 / sqrt(25)) = 3. The z-score can be calculated by subtracting the mean from the desired value and dividing by the standard error. In this case, the z-score is (105 - 100) / 3 = 1.67. Using a standard normal distribution table, we find that the probability is approximately 0.0475.
c. Finding the probability that a randomly selected sample of size n=100 has a mean greater than 105
Using the same process as in part b, we calculate the standard error to be (15 / sqrt(100)) = 1.5. The z-score is then (105 - 100) / 1.5 = 3.33. Using a standard normal distribution table, we find that the probability is approximately 0.0004.
d. Describing the effect of sample size on the probability the mean of a sample will be greater than 105
As the sample size increases, the probability that the mean of the sample will be greater than 105 decreases. This is because as the sample size increases, the standard error decreases, resulting in a smaller z-score and a smaller probability. Therefore, larger sample sizes provide more reliable estimates of the population mean.