Question

The chance that UPS will deliver a package to a customer within X days is normally distributed with a mean of 6 and a standard deviation of 1.3. If there is a 95% chance that the package arrives in X days, how many days will the package take to arrive?

Answer 1

Using the normal distribution properties and the Z-score for a 95% cumulative probability, the UPS package is expected to arrive within approximately 8.14 days.

Step-by-step explanation:

The student is asking about probability in context with a normal distribution. To find the number of days, X, where there is a 95% chance that the UPS package arrives within that time frame, we need to use the properties of the normal distribution. The mean given is 6 days, and the standard deviation is 1.3 days. To find X for the 95% probability, we would look up the z-score that corresponds to the cumulative probability of 0.95 in the standard normal distribution table, and then use the formula X = mean + (z-score * standard deviation). Given that the z-score for 95% in a standard normal distribution is typically around 1.645, we can calculate X as:

X = 6 + (1.645 * 1.3) = 8.1385, which we would round to 8.14 days (since UPS is unlikely to measure delivery time to thousandths of a day).

Note that the exact z-score used could vary slightly depending on the specifics of the standard normal distribution table used, as some tables provide the area to the left of the z-score and would require you to consider the complementary probability (1 - 0.95).