Problem 22 - Calculus for Biology and Medicine

Question

The following data represent a sample from a normal distribution with mean 0 and variance \(1:\) $$ \begin{array}{l} -1.18,0.52,0.36,-0.16,0.92 \\ 0.68,-0.61,-0.54,0.15,1.04 \end{array} $$ Construct a \(95 \%\) confidence interval.

Step-by-Step Solution

Answer

The 95% confidence interval is approximately (-0.501, 0.737).

Step 1: Understand the Data

The data given is a sample from a normal distribution with mean 0 and variance 1. The sample is: \[-1.18, 0.52, 0.36, -0.16, 0.92, 0.68, -0.61, -0.54, 0.15, 1.04\].

Step 2: Calculate the Sample Mean

Sum all sample values and divide by the total number of values to find the sample mean. The sum is \(-1.18 + 0.52 + 0.36 - 0.16 + 0.92 + 0.68 - 0.61 - 0.54 + 0.15 + 1.04 = 1.18\) The sample mean \(\bar{x}\) is \[\bar{x} = \frac{1.18}{10} = 0.118\]

Step 3: Identify the Standard Deviation of Population

Since the variance is given as 1, the standard deviation \(\sigma\) is the square root of the variance: \[\sigma = \sqrt{1} = 1\]

Step 4: Find the Z-Score for 95% Confidence

For a 95% confidence interval in a standard normal distribution, the Z-score is approximately 1.96.

Step 5: Calculate the Margin of Error

The margin of error \(E\) is calculated using the Z-score, standard deviation, and sample size \(n = 10\).\[E = Z \times \left(\frac{\sigma}{\sqrt{n}}\right) = 1.96 \times \left(\frac{1}{\sqrt{10}}\right) \approx 0.619\]

Step 6: Construct the Confidence Interval

The confidence interval is calculated as\[\bar{x} \pm E = 0.118 \pm 0.619\]This results in the interval\[0.118 - 0.619 = -0.501 \] \[0.118 + 0.619 = 0.737 \]So, the confidence interval is approximately \[-0.501, 0.737\].

Key Concepts

Normal Distribution

When dealing with statistics and data analysis, understanding the normal distribution is a key concept. It’s essentially a probability distribution that is symmetric and bell-shaped, where most of the data points cluster around the mean. The normal distribution is also known as the Gaussian distribution.

In our exercise, the data provided comes from a normal distribution with a mean () of 0 and a variance ( ^2) of 1. This specific case is called the standard normal distribution. Here are a few things to remember about the normal distribution:

• 68% of the data falls within 1 standard deviation of the mean.
• 95% of the data falls within 2 standard deviations of the mean.
• 99.7% of the data falls within 3 standard deviations of the mean.

The symmetry of the normal distribution means that the mean, median, and mode are all equal. These properties are used to determine probabilities and calculate confidence intervals, which is part of our solution approach.

Sample Mean

The sample mean is a way of summarizing the data within a sample and provides a good estimate of the population mean. In essence, it is the average of all the data points in the sample. To determine the sample mean, sum up all the individual data points and divide by the total number of points.

In our case, the sample consists of 10 values. By adding all these values together, you get 1.18. Dividing this sum by 10, the number of values, results in a sample mean () of 0.118. This step is crucial because it allows us to estimate the center of our data's distribution.

Understanding the sample mean's role is vital as it serves as the starting point for estimating the entire population's mean, which could be crucial in making broader predictions based on our sample data.

Standard Deviation

The standard deviation is a statistical measure that describes the amount of variability or dispersion within a dataset. In a normal distribution, the standard deviation is key to understanding how much individual data points can differ from the mean. It is calculated as the square root of the variance.

For our dataset, where the variance is 1, the standard deviation (") becomes 1. This indicates that an average data point tends to deviate from the mean by 1 unit.

• A small standard deviation means data points tend to be close to the mean.
• A large standard deviation indicates data points are spread out over a wider range of values.

In probability and statistics, the standard deviation provides insight into the uncertainty and risk associated with a particular measurement or prediction.

Margin of Error

The margin of error gives an idea of the range within which we expect our sample mean to fall in reference to the actual population mean. It assists in building confidence intervals, which tell us how precise our estimate is.

In the calculation, the margin of error () is derived through the Z-score, which corresponds to the confidence level we choose, the standard deviation, and the square root of the sample size. For a 95% confidence level, the Z-score is approximately 1.96. Therefore:

\[ E = 1.96 \times \left(\frac{4}{\sqrt{10}}\right) \approx 0.619 \]

The margin of error shows us an interval within which we can confidently say that the population parameter lies, giving context to statistical estimates. In practice, this helps in reliable decision-making and predicting population trends.