Problem 21 - Calculus for Biology and Medicine

Question

The following data represent a sample from a normal distribution with mean 0 and variance \(1:\) $$ \begin{array}{l} -0.68,1.22,1.33,-0.84,-0.06 \\ 0.50,0.03,-0.13,-0.29,-0.47 \end{array} $$ Construct a \(95 \%\) confidence interval.

Step-by-Step Solution

Answer

The 95% confidence interval is \((-0.4421, 0.5641)\).

Step 1: Calculate the Mean

First, sum up all the data points and divide by the number of data points to find the sample mean. The formula is \( \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i \).Sum: \(-0.68 + 1.22 + 1.33 - 0.84 - 0.06 + 0.50 + 0.03 - 0.13 - 0.29 - 0.47 = 0.61 \).Number of data points \( n = 10 \).Sample mean: \( \bar{x} = \frac{0.61}{10} = 0.061 \).

Step 2: Calculate the Standard Deviation

Use the sample variance formula to find the standard deviation. Sample variance formula is \( s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2 \) and standard deviation \( s = \sqrt{s^2} \).Calculating variance:\[ s^2 = \frac{1}{9}((-0.68-0.061)^2 + (1.22-0.061)^2 + (1.33-0.061)^2 + (-0.84-0.061)^2 + (-0.06-0.061)^2 + (0.50-0.061)^2 + (0.03-0.061)^2 + (-0.13-0.061)^2 + (-0.29-0.061)^2 + (-0.47-0.061)^2) \]After calculations, \( s^2 = 0.6589 \), hence \( s = \sqrt{0.6589} \approx 0.8118 \).

Step 3: Determine the Z-score for 95% Confidence Interval

For a normal distribution, a 95% confidence interval corresponds to a Z-score of approximately 1.96. This value is standard for a two-tailed confidence interval.

Step 4: Calculate the Margin of Error

The formula for the margin of error (E) is \( E = Z \times \frac{s}{\sqrt{n}} \).Substitute the known values:\[ E = 1.96 \times \frac{0.8118}{\sqrt{10}} \approx 1.96 \times 0.2567 \approx 0.5031 \].

Step 5: Construct the Confidence Interval

The confidence interval is given by \( \bar{x} \pm E \).Substitute in the mean and margin of error:\[ 0.061 \pm 0.5031 \]Lower limit: \( 0.061 - 0.5031 = -0.4421 \)Upper limit: \( 0.061 + 0.5031 = 0.5641 \).Therefore, the 95% confidence interval is \((-0.4421, 0.5641)\).

Key Concepts

Standard Deviation

The standard deviation is a measure that tells us how much variation or dispersion exists from the average (mean) value. It essentially shows the spread of a dataset.
In simpler terms, if the data points are close to the mean, the standard deviation will be low; if they are spread out over a wider range, the standard deviation will be high. To calculate the standard deviation for a sample, you first need the sample variance. The formula for sample variance is \[ s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2 \] where \(n\) is the number of data points, \(x_i\) are the individual data points, and \(\bar{x}\) is the sample mean.
The resulting variance is the average of these squared differences. Taking the square root of the variance gives you the standard deviation: \[ s = \sqrt{s^2} \].**Why is Standard Deviation Important?**- **Understanding Variability**: Helps understand data variability. - **Confidence Intervals**: Used in many statistical techniques, including confidence intervals. - **Decision Making**: Assists in data-driven decisions by understanding the spread of the data.

Sample Mean

The sample mean is simply the average of your sample data. It represents a central value for a set of data points, providing a balanced point that considers each piece of data equally. Calculating the sample mean is straightforward: \[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i \] where \(n\) is the number of data points in your sample.
To compute it, add up all the data values and divide by the number of observations.**What makes the Sample Mean beneficial?**- **Center Point**: Offers a measure of central tendency, indicating the middle of the data set. - **Simplicity**: An easy measure to understand and compute. - **Foundation for Analysis**: Forms the basis for further statistical calculations.In the context of a confidence interval, the sample mean helps determine where the central value of a dataset lies and guides you in estimating the true population mean.

Normal Distribution

The normal distribution, often called the bell curve, is a fundamental principle in statistics. It describes a distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. Many natural phenomena approximate a normal distribution. Key features of a normal distribution: - **Symmetry**: The distribution is perfectly symmetrical around its mean. - **Mean = Median = Mode**: In a normal distribution, the mean, median, and mode are all the same value. - **Bell Shape**: The graph of a normal distribution will appear as a "bell curve" **Why is Normal Distribution crucial?** - **Central Limit Theorem**: Suggests that the distribution of the sample mean will approach a normal distribution, regardless of the shape of the population distribution, as the sample size grows. - **Predictive Analysis**: Useful in making predictions and inferential statistics as a wide range of phenomena could be approximated with a normal distribution. - **Standard Comparisons**: Many statistical tests and procedures, such as the confidence interval calculation in this exercise, assume normality of the underlying data. Understanding these principles of normal distribution can considerably enhance your analysis of data behavior and predictability.