Final answer:
a. About 11.9% of the days have more than 1,020 hits. b. About 17.22% of the days have hits between 950 and 1,020. c. If the number of hits is 772 or below, only 15% of the days will have hits below this number.
Step-by-step explanation:
a.
To find the proportion of days with more than 1,020 hits, we need to calculate the z-score and find the area to the right of that value on the standard normal distribution table. First, we calculate the z-score as (1020 - 890) / 110 = 1.18. Looking up the area to the right of 1.18 on the standard normal distribution table, we find approximately 0.1190. So, about 11.9% of the days have more than 1,020 hits.
b.
To find the proportion of days with hits between 950 and 1,020, we need to calculate the z-scores for both values. The z-score for 950 is (950 - 890) / 110 = 0.55, and the z-score for 1,020 is (1020 - 890) / 110 = 1.18. Then, we find the area to the right of 0.55 and subtract the area to the right of 1.18 from it. By looking up these values on the standard normal distribution table, we find that the area to the right of 0.55 is approximately 0.2912 and the area to the right of 1.18 is approximately 0.1190. Subtracting 0.1190 from 0.2912, we get approximately 0.1722. So, about 17.22% of the days have hits between 950 and 1,020.
c.
To find the number of hits such that only 15% of the days will have hits below this number, we need to find the z-score that corresponds to the area of 0.15 to the left of it on the standard normal distribution table. Looking up this value, we find z ≈ -1.04. Now, we can use the z-score formula to solve for the number of hits: (X - 890) / 110 = -1.04. Solving for X, we get X ≈ 772. So, if the number of hits is 772 or below, only 15% of the days will have hits below this number.