Problem 17 - Calculus for Biology and Medicine

Question

Use the random-number generator on a graphing calculator to generate three samples, each of size 10, from a uniform distribution over the interval \((0,1)\). (a) Compute the sample mean and the sample variance of each sample. (b) Combine all three samples, and compute the mean and the sample variance of the combined sample. (c) Compare your answers in (a) and (b) with the true values of the mean and the variance.

Step-by-Step Solution

Answer

Calculated sample means and variances should generally be close to the true mean 0.5 and variance 0.0833, but will vary due to sampling variability.

Step 1: Generate Random Samples

Use the random-number generator on your graphing calculator to generate three separate samples, each containing 10 numbers. Make sure each number is drawn from a uniform distribution ranging from 0 to 1.

Step 2: Calculate Sample Means

For each of the three samples generated, sum all the numbers and divide by the sample size (10) to find the sample mean. Let these means be denoted as \( \bar{x}_1, \bar{x}_2, \bar{x}_3 \) for Sample 1, Sample 2, and Sample 3, respectively.

Step 3: Calculate Sample Variances

For each sample, compute the variance by finding the squared differences between each number and the sample mean, summing them all up, and then dividing by the sample size minus one (9 for each sample). Denote these variances as \( s_1^2, s_2^2, s_3^2 \).

Step 4: Combine Samples

Combine all three samples into a single sample containing 30 numbers. This involves merging the lists generated for each individual sample.

Step 5: Calculate Combined Sample Mean

Sum all the numbers in the combined sample (all 30 numbers) and then divide by 30 to compute the mean of the combined sample. This gives the combined sample mean \( \bar{x}_{combined} \).

Step 6: Calculate Combined Sample Variance

Compute the variance for the combined sample by finding the squared differences between each number in the combined sample and the combined sample mean, sum all these squared differences, and divide by 29 (n-1 for the combined sample's n=30). Denote this as \( s_{combined}^2 \).

Step 7: Compare with True Mean and Variance

Recall that for a uniform distribution over the interval \((0,1)\), the true mean is \( \mu = 0.5 \) and the true variance is \( \sigma^2 = \frac{1}{12} \approx 0.0833 \). Compare the calculated means and variances from steps 2, 3, 5, and 6 with these true values.

Key Concepts

Uniform Distribution

A uniform distribution is a type of probability distribution in which every outcome is equally likely to occur within a defined range. In our exercise, we are dealing with a uniform distribution over the interval \((0,1)\).
This means that every number between 0 and 1 is equally probable when randomly selected. The defining characteristics of a uniform distribution in this interval are:

• Mean: For a continuous uniform distribution from \((a,b)\), the mean \(\mu\) is calculated as \(\frac{a+b}{2}\). Therefore, in our interval, it is \(0.5\).
• Variance: The variance \(\sigma^2\) is given by \(\frac{(b-a)^2}{12}\). Thus, here it is approximately 0.0833.

Understanding these properties helps us compare our calculated sample statistics against the expected values from a perfectly uniform distribution.

Sample Mean

The sample mean is a fundamental statistical measure that is the average of all the observations in a sample. In our exercise, each of the three samples has a size of 10. To calculate the sample mean \(\bar{x}\), sum all the individual numbers in a sample and then divide by the sample size. For example:

• If a sample consists of the numbers \((x_1, x_2, ..., x_{10})\), the sample mean is \(\bar{x} = \frac{x_1 + x_2 + ... + x_{10}}{10}\).

This statistic provides a central value of the dataset, giving insight into the general 'center' of the data sampled from our uniform distribution.
Comparing each sample mean with the theoretical mean of 0.5 allows us to see how typical or atypical our sample is from what we would expect in an ideal uniform distribution.

Sample Variance

Sample variance is a measure of how much the values in a sample deviate from their mean. It's a critical statistic for understanding the dispersion of data points. For each sample, calculate the variance by:

• First finding the squared difference between each data point and the sample mean \((x_i - \bar{x})^2\).
• Sum all these squared differences.
• Then divide this sum by the sample size minus 1, which is 9 in our case, to account for 'degrees of freedom'.

The formula is: \[s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}\],where \(n = 10\).
Comparing these with the theoretical variance of approximately 0.0833 helps us understand the typical spread we should expect if our data were drawn perfectly from a uniform distribution.

Graphing Calculator

A graphing calculator is a powerful tool for statistics and mathematics. It enables fast computations, facilitating both learning and application of complex concepts such as random sampling. In this exercise, it served a critical role in:

• Generating random samples from a uniform distribution over the interval \((0,1)\) using its built-in random-number generator.
  This capability ensures that we practice with real, random data.
• Performing calculations for sample means and variances.
  While these can be done manually, a graphing calculator speeds up the process, allowing us more time to analyze and interpret results.

Ensuring familiarity and proficiency with such tools enhances our understanding and application of statistical methods in various fields.

Statistics in Biology

Statistics is pivotal in biology as it enables researchers to decipher patterns, relationships, and trends in biological data. Understanding tools like uniform distribution, sample mean, and sample variance can greatly aid in biological research. For example, consider a case:

• Ecologists might use random sampling to estimate species abundance in a specific area. A uniform distribution could model the presence probability of organisms across different habitats.
• Sample means can help geneticists determine average trait values in a population, like average height or weight.
• Sample variance informs epidemiologists about diversity in disease prevalence across different regions.

Applying statistics judiciously aids in making sound scientific conclusions, demonstrating its indispensability in answering biological inquiries.