It takes the bus an average of 20 minutes to arrive at a certain bus stop. A student is waiting for the next
bus to arrive.
a) What is the probability the student waits more than 30 minutes?
b) What is the probability the student waits exactly 15 minutes?
c) What is the probability the student waits more than 45 minutes (total) given they've already waited
20 minutes?
d) Given the student waited less than 45 minutes, what is the probability they waited less than 20
minutes?
e) What is the standard deviation of waiting time?
f) What is the median waiting time?
Inadequate data.
I think there is enough actually
To answer these probability questions, we need to assume that the arrival times of the buses at the bus stop follow a certain distribution. Let's assume that the arrival times are normally distributed with a mean of 20 minutes and a standard deviation of σ (unknown for now).
Now let's go through each question and explain how to calculate the probabilities:
a) What is the probability the student waits more than 30 minutes?
To calculate this probability, we need to find the area under the normal distribution curve to the right of 30 minutes. We can use a standard normal distribution table or a calculator to find the corresponding z-score for 30 minutes and then find the area under the curve to the right of that z-score.
b) What is the probability the student waits exactly 15 minutes?
Since the arrival times are assumed to be continuous and normally distributed, the probability of exactly 15 minutes is technically zero. In practice, we often approximate such probabilities as zero for continuous distributions.
c) What is the probability the student waits more than 45 minutes (total) given they've already waited 20 minutes?
To calculate this conditional probability, we need to find the area under the normal distribution curve to the right of 45 minutes, given that we know the student has already waited 20 minutes. This is equivalent to finding the probability of waiting more than 25 minutes (45 - 20), which we can calculate using similar steps as in question a).
d) Given the student waited less than 45 minutes, what is the probability they waited less than 20 minutes?
This question asks for a conditional probability. We need to find the area under the normal distribution curve to the left of 20 minutes, given that we know the student has waited less than 45 minutes. This is equivalent to finding the probability of waiting less than 20 minutes, which is a straightforward calculation using the normal distribution.
e) What is the standard deviation of waiting time?
The standard deviation is not given, so we cannot calculate it directly. If we have a larger dataset of waiting times, we can estimate the standard deviation using statistical methods. However, if we assume the distribution of waiting times follows a normal distribution, the standard deviation is a parameter that needs to be known separately.
f) What is the median waiting time?
The median waiting time represents the value that separates the data into two equal halves. In a normal distribution, the median is also equal to the mean. Therefore, the median waiting time is 20 minutes, as given in the problem statement.
To summarize, some of the probabilities can be calculated directly using the normal distribution properties, given the relevant parameters, while others require additional information or assumptions. The standard deviation can be estimated from a dataset, but we need more information to determine it in this specific case. The median, on the other hand, is equal to the mean in a normal distribution, so it is given as 20 minutes.