For this problem, we are given the mean and standard deviation of a certain test. We need to determine a probability of a random sample to be in a few values.
The first value we need to determine is the probability of the random sample being below 70. The first step we need to take is to determine the z-score of this value, which can be calculated with the following expression:
For the value of 70, we have:
Now we need to find this value on the z-table, which is:
Therefore we can determine that the probability of a value to be below 70 is 6.68%.
Now we need to determine the probability of a value above 112. We need to determine the z-score once again:
The z-table only tells us values below the z-score, so we need to subtract the result from 1, which is shown below:
The probability of the value being greater than 112 is 27.25%.
Now we need to find the probability of the score to be between 85 and 115. We need to find both Z-scores:
So we need to find the two values on the Z-table and subtract them. We have:
[tex]P(-0.75The probability of the random value being between 85 and 115 is 54.68%.