Final answer:
To find the z-score a, such that P (0 < z < a) = 0.3106, we look up the cumulative area of 0.8106 (0.5+0.3106) in the z-table and find a to be approximately 0.88.
Step-by-step explanation:
The question involves using the standard normal distribution to find the value of a z-score, a, given that P (0 < z < a) = 0.3106.
Since the standard normal distribution is symmetric about the mean, and the mean is 0, we want to find the value of a such that the area between z = 0 and z = a equals 0.3106.
To find a, we consult z-tables that provide the cumulative probability (area under the curve) to the left of any given z-score.
Since the z-table gives us the area to the left of any z-score, we need to find the z-score for which the area to the left is 0.5 (the area to the left of z=0) plus 0.3106 (the area between z=0 and z=a), which sums to 0.8106.
By looking up this cumulative area in the z-table, we can find the corresponding z-score, which will be our value for a.
Upon checking a standard z-table, we find that a z-score of approximately 0.88 corresponds to a cumulative area of 0.8106. Therefore, a ≈ 0.88.