The times taken to complete a statistics test by all students are normally distributed with a mean of 120 minutes and a standard deviation of 12 minutes. Find the probability that the mean time taken to complete this test by a random sample of 16 students would be between 122 and 125 minutes.
To find the probability that the mean time taken to complete the test by a random sample of 16 students is between 122 and 125 minutes, we can use the Central Limit Theorem.
The Central Limit Theorem states that for large sample sizes, the distribution of the sample means will approach a normal distribution, regardless of the shape of the population distribution.
In this case, the population distribution is normally distributed with a mean of 120 minutes and a standard deviation of 12 minutes.
The mean of the sample means, also known as the sample mean distribution, will still be 120 minutes.
However, the standard deviation of the sample means (also known as the standard error) will be the population standard deviation divided by the square root of the sample size. In this case, the sample size is 16.
Standard error = population standard deviation / √sample size = 12 / √16 = 12 / 4 = 3
Now we can calculate the z-scores for the upper and lower bounds of the mean time:
Lower z-score = (122 - 120) / 3 = 2 / 3 ≈ 0.67
Upper z-score = (125 - 120) / 3 = 5 / 3 ≈ 1.67
Next, we need to calculate the probabilities associated with these z-scores using a standard normal distribution table or calculator.
Using the table or calculator, we find that the probability corresponding to the lower z-score is approximately 0.7486, and the probability corresponding to the upper z-score is approximately 0.9525.
To find the probability that the mean time taken to complete the test is between 122 and 125 minutes, we subtract the lower probability from the upper probability:
P(122 ≤ X ≤ 125) = P(X ≤ 125) - P(X ≤ 122) = 0.9525 - 0.7486 = 0.2039
Therefore, the probability that the mean time taken to complete the test by a random sample of 16 students would be between 122 and 125 minutes is approximately 0.2039, or 20.39%.