The number of vacation days taken by employees of a company is normally distributed with a mean of 14 days and a standard deviation of 3 days. For the next employee, what is the probability that the number of days of vacation taken is less than 10 days? More than 21 days?
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to each Z score.
To find the probability that the number of days of vacation taken by the next employee is less than 10 days, we need to calculate the z-score corresponding to 10 days and then use the z-table to find the corresponding probability. Similarly, to find the probability that the number of days taken is more than 21 days, we need to calculate the z-score corresponding to 21 days and then use the z-table to find the probability of being greater than that value.
First, let's calculate the z-scores:
Z-score (less than 10 days) = (10 - 14) / 3 = -1.33
Z-score (more than 21 days) = (21 - 14) / 3 = 2.33
Now we can use the z-table, which provides the probability for a given z-score. The z-table gives probabilities for values up to a certain z-score. Since we need to find the probability of being less than 10 days, we need to find the cumulative probability for the z-score of -1.33.
Looking up the z-score of -1.33 in the z-table, we find that the corresponding cumulative probability is approximately 0.0918. This means that there is a 9.18% chance that the number of days taken by the next employee is less than 10 days.
Similarly, to find the probability of taking more than 21 days, we need to find the cumulative probability for the z-score of 2.33. Looking up this z-score in the z-table, we find that the corresponding cumulative probability is approximately 0.9902. This means that there is a 99.02% chance that the number of days taken by the next employee is more than 21 days.
So, to summarize:
- The probability that the number of days taken is less than 10 is approximately 9.18%
- The probability that the number of days taken is more than 21 is approximately 99.02%