91Ó°ÊÓ

A pharmaceutical company wishes to know whether an experimental drug being tested in its laboratories has any effect on systolic blood pressure. Fifteen randomly selected subjects were given the drug, and their systolic blood pressures (in millimeters) are recorded. \(\begin{array}{lll}172 & 148 & 123\end{array}\) \(\begin{array}{lll}140 & 108 & 152\end{array}\) \(\begin{array}{lll}123 & 129 & 133\end{array}\) \(\begin{array}{lll}130 & 137 & 128\end{array}\) \(\begin{array}{lll}115 & 161 & 142\end{array}\) a. Guess the value of \(s\) using the range approximation. b. Calculate \(\bar{x}\) and \(s\) for the 15 blood pressures. c. Find two values, \(a\) and \(b\), such that at least \(75 \%\) of the measurements fall between \(a\) and \(b\).

Step by step solution

01

Organize the data and find the range

In order to guess the value of the standard deviation, we first need to find the range of the data, which is the difference between the highest and lowest data points. Let's organize the data in ascending order: \(108, 115, 123, 123, 128, 129, 130, 133, 137, 140, 142, 148, 152, 161, 172\) Now let's find the range of the data: \(Range = 172 - 108 = 64\)

02

Guess the standard deviation using the range approximation

Since we have the range, we can now guess the value of the standard deviation using the range approximation. The range approximation states that the standard deviation (\(s\)) can be approximated by dividing the range by 4: \(s \approx \frac{Range}{4} = \frac{64}{4} = 16\) So our guess for the standard deviation is 16.

03

Calculate the mean of the data

To calculate the mean (\(\bar{x}\)) of the systolic blood pressure measurements, we add all the values and divide by the number of subjects (15): \(\bar{x} = \frac{108+115+123+123+128+129+130+133+137+140+142+148+152+161+172}{15} = \frac{2019}{15} \approx 134.6\) So the mean systolic blood pressure is approximately 134.6 millimeters.

04

Calculate the standard deviation of the data

To calculate the standard deviation (\(s\)), we first need to find the sum of the squared differences between each data point and the mean: \(\sum (x_i - \bar{x})^2 = (108-134.6)^2 + (115-134.6)^2 +...(172-134.6)^2 \approx 11924.4\) Next, divide the result by the number of subjects minus 1 (15-1=14) and take the square root of the result: \(s = \sqrt{\frac{11924.4}{14}} \approx 29.2\) So the calculated standard deviation is approximately 29.2 millimeters.

05

Find two values, a and b, such that at least 75% of the measurements fall between a and b

To find two values, a and b, such that at least 75% of the measurements fall between a and b, we can use Chebyshev's inequality. Chebyshev's inequality states that for any given data set, at least 75% of the data will fall within two standard deviations of the mean: \(a = \bar{x} - 2s \approx 134.6 - 2(29.2) \approx 76.2\) \(b = \bar{x} + 2s \approx 134.6 + 2(29.2) \approx 193\) Therefore, at least 75% of the measurements fall between 76.2 millimeters and 193 millimeters.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standard Deviation

Understanding standard deviation is crucial when analyzing data because it helps us see how much variation there is from the average (mean) of the data set. The standard deviation, noted as \(s\), tells us how spread out the numbers are in a data set. If the standard deviation is low, the data points tend to be close to the mean. If it is high, the data points are more spread out.To find the standard deviation:

• First, calculate the mean of the data.
• Subtract the mean from each data point and square the result.
• Add all those squared differences.
• Divide this sum by the number of data points minus one (for a sample), which gives the variance.
• Finally, take the square root of the variance to find the standard deviation.

This step-by-step approach ensures that you accurately gauge the variability in your data set.

Mean Calculation

The mean, often referred to as the average, is a fundamental concept in statistics to find the central tendency of a data set. Calculating the mean provides a simple summary of data points, which helps in comparing different data sets or observing changes over time.To calculate the mean:\( \bar{x} \):

• Add together all the data points.
• Divide the sum by the number of data points in the set.

For instance, if you have 15 systolic blood pressure readings, sum them all and divide by 15 to find the mean. This method brings everyone to a central idea of what the typical value might be in a set, though keep in mind that it may not represent spectacular peaks or drops in data.

Chebyshev's Inequality

Chebyshev's inequality is a useful statistical tool that provides bounds on how much data lies within a certain number of standard deviations from the mean, applicable to any data distribution. This is particularly helpful when you do not assume the data follows a normal distribution.For any data set, Chebyshev's inequality states that the proportion of results within \(k\) standard deviations of the mean is at least \(1 - \frac{1}{k^2}\). For example, setting \(k = 2\):

• Calculate \( \bar{x} - 2s \) and \( \bar{x} + 2s \) to determine the range in which at least 75% of your data points will fall.
• This ensures that outliers do not detract from an understanding of the general trend.

In the example of the pharmaceutical analysis of blood pressure, using Chebyshev's inequality helps determine how many subjects' pressures fall in a typical range, contributing to understanding the effect of the drug tested.