Question

The length of time required for an automobile driver to respond to a particular emergency situation was recorded for \(n=10\) drivers. The times (in seconds) were \(.5, .8,1.1, .7, .6,\) .9, .7, .8, .7, .8 a. Scan the data and use the procedure in Section 2.5 to find an approximate value for \(s\). Use this value to check your calculations in part b. b. Calculate the sample mean \(\bar{x}\) and the standard deviation \(s\). Compare with part a.

Short answer

Answer: Based on our calculations, the sample mean (x̅) is 0.76, and the standard deviation (s) is approximately 0.228.

Step-by-step solution

Analyze the data

We are given the following response times (in seconds) for 10 drivers: .5, .8, 1.1, .7, .6, .9, .7, .8, .7, .8.

Use procedure in Section 2.5 to find an approximate value for s

Unfortunately, we don't have access to Section 2.5, but we can still calculate the standard deviation using the formula.

Calculate the sample mean x̅

To calculate x̅, we add up all the values and divide by the total number of values (10): \(x̅ = \frac{(.5 + .8 + 1.1 + .7 + .6 + .9 + .7 + .8 + .7 + .8)}{10} = \frac{7.6}{10}=0.76\)

Calculate the sample standard deviation s

The formula for the standard deviation is: \(s = \sqrt{\frac{1}{n-1}\sum_{i=1}^n (x_i -\bar{x})^2}\) So, we need to calculate the sum of squared differences between each value and the sample mean: \((0.5-0.76)^2+(0.8-0.76)^2+(1.1-0.76)^2+(0.7-0.76)^2+(0.6-0.76)^2+(0.9-0.76)^2+(0.7-0.76)^2+(0.8-0.76)^2+(0.7-0.76)^2+(0.8-0.76)^2=0.468\) Now, we can plug the values into the standard deviation formula: \(s = \sqrt{\frac{1}{10-1}(0.468)}=\sqrt{\frac{1}{9}(0.468)}=\sqrt{0.052}=0.228\) So, the sample standard deviation is approximately 0.228.

Compare the results

Since we don't have the simplified version of calculating standard deviation from the mentioned Section 2.5, we can only provide the exact sample mean and standard deviation calculated in steps 3 and 4 which are 0.76 and 0.228 respectively.

Key concepts

Descriptive Statistics

Descriptive statistics provide a summary of data through numbers, graphs, and tables. They simplify large amounts of data into manageable chunks, aiding in the visualization and understanding of a dataset's essential features. This form of statistics includes several measures: central tendency (mean, median, and mode), variability (range, variance, standard deviation), and distribution shape (skewness, kurtosis).

For instance, when you have a list of times like the response times of drivers in our exercise, descriptive statistics will help identify the average time, the spread of times around this average, and other characteristics that help researchers understand driving behaviors in emergency situations. It's a fundamental tool for researchers, analysts, and students alike to summarize and describe data efficiently.

Mean Calculation

The mean, often referred to as the average, is calculated by summing up all the values in a dataset and dividing by the number of values. The formula for the sample mean is represented as \( \bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i \) where \( x_i \) are the values and \( n \) is the total number of values.

In our exercise, the mean response time of the 10 drivers was calculated as 0.76 seconds. This indicates that if we took a simple average of the drivers' response times, a typical driver would take approximately 0.76 seconds to respond to an emergency. The calculation of the mean is a fundamental step since it's often used as a reference point for other statistical analyses, such as the calculation of variance and standard deviation.

Variance and Standard Deviation

Variance and standard deviation are both measures of variability within a dataset. Variance is the average of the squared differences from the mean, giving you an idea of the spread of data points. The formula for the sample variance is \( s^2 = \frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2 \), where \( s^2 \) is the sample variance, \( x_i \) are the data points, \( \bar{x} \) is the mean, and \( n \) is the number of data points.

Standard deviation is the square root of the variance and provides a measure of the spread of data points around the mean that is in the same unit of measurement as the data. It is represented by the formula \( s = \sqrt{s^2} \). A small standard deviation indicates that the data points are close to the mean, while a large standard deviation indicates that the data points are spread out over a wider range of values. In our exercise, the standard deviation of response times among 10 drivers was calculated as approximately 0.228 seconds, telling us that on average, individual response times were roughly 0.228 seconds away from the mean response time.