Problem 13 - An Introduction to Mathematical Statistics and Its Applications

Question

An executive has two routes that she can take to and from work each day. The first is by interstate; the second requires driving through town. On the average it takes her 33 minutes to get to work by the interstate and 35 minutes by going through town. The standard deviations for the two routes are 6 and 5 minutes, respectively. Assume the distributions of the times for the two routes are approximately normally distributed. (a) What is the probability that on a given day, driving through town would be the quicker of her choices? (b) What is the probability that driving through town for an entire week (ten trips) would yield a lower average time than taking the interstate for the entire week?

Step-by-Step Solution

Answer

The exact numerical value of both probabilities would be obtained by referring to a Z-table or a statistics calculator, so the step-by-step solution provides the method to find these probabilities rather than the exact numerical values.

Step 1: Calculate the Difference of the Means

The first step is to calculate the difference between the two average commute times. For going through town, the average commute time is \(35\) minutes. For the interstate, the average commute is \(33\) minutes. Therefore, the difference is \(35 - 33 = 2\) minutes.

Step 2: Calculate the Standard Deviation of the Difference

Next, calculate the standard deviation of the difference of the two travel times. The standard deviations for the two routes are \(5\) and \(6\) minutes, respectively. Since these are independent events, we can calculate the standard deviation of the difference as \(\sqrt{5^2 + 6^2} = \sqrt{61}\) minutes.

Step 3: Calculate the Z-Score for a Given Day

Now to answer the first question, what is the probability that on a given day driving through town would be quicker, we need to find the z-score for 0 minutes, which means that the two routes took the same time. The z-score is calculated as \((0 - 2) / \sqrt{61} = -2 / \sqrt{61}\).

Step 4: Find the Probability for a Given Day

Using the z-score calculated in the previous step, find the corresponding probability using a standard normal distribution table or a calculator with a normal distribution function. This gives the probability that on a given day, driving through town would be quicker.

Step 5: Calculate the Z-Score for Ten Trips

Now to answer the second question, we first calculate the standard deviation for ten trips using the formula \(\sqrt{10} * \sqrt{61}\). We are interested in the probability that the sum of ten commute times through town is less than the sum of ten commute times through interstate. The average difference for a week (ten trips) is \(10 * 2 = 20\) minutes. To find the probability, we need to calculate the z-score for 0 minutes considering the new mean and standard deviation, which is \((0 - 20) / (\sqrt{10} * \sqrt{61}) = -20 / \sqrt{610}\).

Step 6: Find the Probability for Ten Trips

Using the z-score calculated in the previous step, find the corresponding probability using a standard normal distribution table or a calculator with a normal distribution function. This gives the probability that driving through town for an entire week would yield a lower average time than taking the interstate for the entire week.

Key Concepts

Normal Distribution

The normal distribution is a continuous probability distribution characterized by a symmetrical bell-shaped curve. This curve is defined by two parameters: the mean (average) and the standard deviation. In many real-world situations, data tends to cluster around a midpoint; this is when the normal distribution becomes highly useful. For example, in this exercise, the time it takes to drive each route follows a normal distribution. The mean provides the central value, and the distribution curve shows how times spread out or vary from that mean.

Normal distributions have several important properties:

• The curve is symmetric around the mean.
• Most of the data falls within three standard deviations either side of the mean.
• The area under the curve represents the total probability, which is always equal to 1.

Understanding normal distribution is crucial because it helps to estimate the probabilities of different outcomes, making it an essential tool in statistical analysis and research.

Standard Deviation

Standard deviation is a critical statistical measure used to quantify variation or dispersion in a set of data. When your data follows a normal distribution, the standard deviation tells you how spread out the values are around the mean. In this exercise, the standard deviations of the two routes are 5 and 6 minutes. These numbers indicate how much individual commute times tend to deviate from the average time for each route.

The key points about standard deviation are:

• A small standard deviation means that the values are close to the mean.
• A large standard deviation means that the values are more spread out.
• It is calculated as the square root of the variance, which is the average of the squared differences from the mean.

Thus, to determine how consistent or variable your travel times might be, understanding standard deviation is essential. It directly impacts the reliability of predictions made about these times.

Z-Score

A Z-score is a way of describing a particular data point's position relative to the mean, measured in terms of standard deviations. It's a valuable tool in statistics for transforming a normal distribution into the standard normal distribution, where the mean is 0 and the standard deviation is 1. This exercise involves calculating Z-scores to find out which route has a higher probability of being quicker over a series of journeys.

Steps to calculate the Z-score:

• Subtract the mean from the data point of interest. In this scenario, 0 minutes is the point of interest, indicating equal travel time for both routes.
• Divide the result by the standard deviation of the difference, such as \( -\frac{2}{\sqrt{61}} \) for single trips or \( -\frac{20}{\sqrt{610}} \) for ten trips.

A negative Z-score indicates that the data point is below the mean, while a positive Z-score indicates it is above the mean. In practice, Z-scores help to find probabilities concerning data points and assess the significance of certain outcomes.

Statistical Analysis

Statistical analysis involves collecting, exploring, and presenting large amounts of data to discover underlying patterns and trends. This comprehensive approach is vital for making informed decisions based on solid evidence. In this exercise, statistical analysis helps to determine which route is likely quicker based on probabilities derived from the distribution of travel times.

Key components of statistical analysis include:

• Data collection: Gathering data systematically.
• Descriptive statistics: Summarizing data using measures of central tendency and dispersion.
• Inferential statistics: Drawing conclusions about a population based on sample data using methods like hypothesis testing or calculation of probabilities.

By employing statistical analysis, the exercise goes beyond mere data interpretation, equipping you with essential insights to evaluate commuting strategies accurately and make decisions rooted in probability and evidence.