Problem 49 - Basic Statistics for Business and Economics with Student

Question

Police Chief Edward Wilkin of River City reports 500 traffic citations were issued last month. A sample of 35 of these citations showed the mean amount of the fine was \(\$ 54,\) with a standard deviation of \(\$ 4.50 .\) Construct a 95 percent confidence interval for the mean amount of a citation in River City.

Step-by-Step Solution

Answer

The confidence interval is \( (\$52.46, \$55.54) \).

Step 1: Determine the Sample Statistics

Identify the known sample statistics from the problem: the sample size \( n = 35 \), the sample mean \( \bar{x} = 54 \), and the sample standard deviation \( s = 4.50 \).

Step 2: Identify the Confidence Level and Corresponding t-Value

We are asked to construct a 95% confidence interval. Since the sample size is\( n = 35 \) (which is less than 30), we need to use the t-distribution for more accurate results. The degrees of freedom (df) is \( n - 1 = 34 \). For a 95% confidence interval and \( df = 34 \), the t-value (from t-table) is approximately 2.032.

Step 3: Calculate the Standard Error of the Mean (SEM)

The standard error of the mean is calculated using the formula:\[SEM = \frac{s}{\sqrt{n}}\]Substitute the known values:\[SEM = \frac{4.50}{\sqrt{35}} \approx 0.76\]

Step 4: Construct the Confidence Interval

Use the formula for the confidence interval of the mean:\[\bar{x} \pm (t \times SEM)\]Substitute the known values:\[ CI = 54 \pm (2.032 \times 0.76) \]Calculate the margin of error:\[ ME = 2.032 \times 0.76 \approx 1.54 \]Thus, the confidence interval is:\[ (54 - 1.54, 54 + 1.54) = (52.46, 55.54) \]

Step 5: Conclusion

The 95% confidence interval for the mean amount of a traffic citation in River City is from \( \\(52.46 \) to \( \\)55.54 \). This means we are 95% confident that the true mean fine lies within this range.

Key Concepts

Sample Statistics

Sample statistics are the backbone of data analysis as they help us summarize information from a sample.
In the given exercise, we deal with a sample of traffic citations drawn from a larger population in River City.Key Components of Sample Statistics:

• Sample Size (): It refers to the number of observations in the sample. In our case, the sample size is 35.
• Sample Mean (\( \bar{x} \)): This is the average value of the sample, which provides an estimate of the population mean. Here, the sample mean is \\(54.
• Sample Standard Deviation (\( s \)): It measures the amount of variation or dispersion of the sample values. In this exercise, it is \\)4.50.

Sample statistics help us make inferences about the population.
By using the sample mean, we can estimate what the average traffic fine might be for the entire city.

Standard Error

Standard Error (SE) is a critical concept for understanding how much our sample mean estimate might vary if we took numerous samples from the same population.
It provides insight into the precision of our sample mean as an estimate of the true population mean.Calculating the Standard Error:The Standard Error of the Mean (SEM) is found using the formula:\[SEM = \frac{s}{\sqrt{n}}\]Where:

• \( s \) is the sample standard deviation (\$4.50 here).
• \( n \) is the sample size (35 in our example).

Substituting the known values, the SEM is approximately \(0.76\).
A smaller SEM indicates that the sample mean is a more accurate reflection of the population mean.

t-Distribution

The t-distribution plays a crucial role when working with small sample sizes, especially when the sample size is less than 30.
Unlike the normal distribution, the t-distribution is broader and has heavier tails, which accounts for the additional variability seen in smaller samples.Understanding the Use of t-Distribution:

• Degrees of Freedom (df): Indicates the number of values that are free to vary in the calculation of a statistic. For a sample, it is calculated as \( n - 1 \). In this exercise, df is \(34\).
• Confidence Level: The probability that the calculated interval contains the true population parameter. We are using a 95% confidence level.
• t-Value: Derived from the t-distribution table based on the degrees of freedom and confidence level. Here, the t-value is approximately 2.032.

The t-distribution enables us to construct a reliable confidence interval even with a smaller sample.
It accounts for extra variability and ensures more accurate predictions of the population parameters.