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Ch. 7 - Estimating Parameters and Determining Sample Sizes
Triola - Elementary Statistics 14th Edition
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Tutti i libri di testoTriola 14th EditionCh. 7 - Estimating Parameters and Determining Sample SizesProblema 7.2.17
Capitolo 7, Problema 7.2.17

Genes Samples of DNA are collected, and the four DNA bases of A, G, C, and T are coded as 1, 2, 3, and 4, respectively. The results are listed below. Construct a 95% confidence interval estimate of the mean. What is the practical use of the confidence interval?


2 2 1 4 3 3 3 3 4 1

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Step 1: Calculate the sample mean (\( \bar{x} \)) of the given data. Add all the values in the dataset (2, 2, 1, 4, 3, 3, 3, 3, 4, 1) and divide by the total number of observations (n = 10). The formula is \( \bar{x} = \frac{\sum x_i}{n} \).
Step 2: Calculate the sample standard deviation (s). Use the formula \( s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \), where \( x_i \) represents each data point, \( \bar{x} \) is the sample mean, and \( n \) is the sample size.
Step 3: Determine the critical value (t*) for a 95% confidence level. Since the sample size is small (n = 10), use the t-distribution table with degrees of freedom \( df = n - 1 \) (in this case, \( df = 9 \)). Look up the t-value corresponding to a 95% confidence level.
Step 4: Calculate the margin of error (ME) using the formula \( ME = t^* \cdot \frac{s}{\sqrt{n}} \), where \( t^* \) is the critical value, \( s \) is the sample standard deviation, and \( n \) is the sample size.
Step 5: Construct the confidence interval. The 95% confidence interval is given by \( \bar{x} \pm ME \), where \( \bar{x} \) is the sample mean and \( ME \) is the margin of error. Interpret the interval in the context of the problem, explaining that it provides a range of plausible values for the true mean of the DNA base codes.

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Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter with a specified level of confidence, typically 95%. It provides an estimate of uncertainty around the sample mean, indicating how much the sample mean might vary from the actual population mean.
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06:33
Introduction to Confidence Intervals

Sample Mean

The sample mean is the average of a set of values collected from a sample, calculated by summing all the sample values and dividing by the number of observations. It serves as a point estimate of the population mean and is crucial for constructing confidence intervals.
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05:11
Sampling Distribution of Sample Proportion

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. In the context of confidence intervals, it helps quantify the variability of the sample data, which is essential for determining the width of the confidence interval and thus the precision of the estimate.
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Calculating Standard Deviation
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Confidence Levels

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