Problem 14 - Basic Statistics for Business and Economics with Student

Question

14\. The Greater Pittsburgh Area Chamber of Commerce wants to estimate the mean time workers who are employed in the downtown area spend getting to work. A sample of 15 workers reveals the following number of minutes spent traveling. \begin{tabular}{|llllllll|} \hline 29 & 38 & 38 & 33 & 38 & 21 & 45 & 34 \\ 40 & 37 & 37 & 42 & 30 & 29 & 35 & \\ \hline \end{tabular} Develop a 98 percent confidence interval for the population mean. Interpret the result.

Step-by-Step Solution

Answer

We are 98% confident that the mean travel time is between 30.65 and 39.49 minutes.

Step 1: Calculate the Sample Mean

First, add up all the individual travel times from the sample: \(29, 38, 38, 33, 38, 21, 45, 34, 40, 37, 37, 42, 30, 29, 35.\) The sum is \(526\). Next, divide this sum by the number of observations, which is \(15\), to find the sample mean \(\bar{x}\). So, \[ \bar{x} = \frac{526}{15} = 35.067. \] Thus, the sample mean is approximately \(35.07\) minutes.

Step 2: Calculate the Sample Standard Deviation

To find the sample standard deviation, first calculate the squared deviations of each sample value from the mean \(\bar{x} = 35.067\). Then, sum these squared deviations and divide by the sample size minus one, \(n-1 = 14\), and finally take the square root. After calculating, the sample standard deviation \(s\) is found to be approximately \(6.53\) minutes.

Step 3: Determine the t-Value for the Confidence Interval

Since we are looking for a 98% confidence interval with a sample size of \(15\), which means \(14\) degrees of freedom, use the t-distribution table to find the critical value \(t_{\alpha/2}\). For a 98% confidence level and \(14\) degrees of freedom, \(t_{\alpha/2} \approx 2.624\).

Step 4: Calculate the Margin of Error

The margin of error (E) is calculated using the formula \[ E = t_{\alpha/2} \times \frac{s}{\sqrt{n}}. \] Substitute the values: \(t_{\alpha/2} = 2.624\), \(s = 6.53\), and \(n = 15\). Therefore, \[ E = 2.624 \times \frac{6.53}{\sqrt{15}} \approx 4.42. \]

Step 5: Develop the Confidence Interval

With the sample mean \(\bar{x} = 35.067\) and the margin of error \(E = 4.42\), the 98% confidence interval is constructed as: \[ (\bar{x} - E, \bar{x} + E) = (35.067 - 4.42, 35.067 + 4.42) = (30.65, 39.49). \]

Step 6: Interpret the Result

We are 98% confident that the true mean travel time for workers in the downtown area is between \(30.65\) and \(39.49\) minutes. This interval gives us a range in which the actual average travel time is likely to fall.

Key Concepts

Sample Mean

The sample mean is an important concept when analyzing datasets. It represents the average of the data points and provides a central tendency for the sample.
In the context of our exercise, we collected travel times from 15 workers. By adding these times together and dividing by the number of observations, we found the sample mean.
Here's how it works:

• Add all data points: 29, 38, 38, 33, 38, 21, 45, 34, 40, 37, 37, 42, 30, 29, 35 to get a total of 526.
• Divide this sum by the number of data points, which is 15.
• So, the sample mean is about 35.07 minutes.

The sample mean is an estimate of the population mean, helping us to understand the average time workers spend commuting.

T-Distribution

The t-distribution is a key concept, especially when dealing with small sample sizes. It is similar to the normal distribution but has heavier tails, meaning it is more prone to produce values that fall far from its mean. This property makes it very useful when estimating variances of small samples.
In this exercise, we used the t-distribution to find a critical value, which helped us construct the confidence interval:

• Because our sample size was 15, we used the t-distribution with 14 degrees of freedom (one less than the sample size).
• For a confidence level of 98%, we found the t-value from the t-distribution table, which was approximately 2.624.

Using the t-distribution allows us to build more accurate confidence intervals when working with small samples.

Margin of Error

The margin of error gives us a range in which the true population mean is likely to fall. It accounts for the variability and uncertainty inherent in a sample.
Here's how the margin of error is calculated in our example:

• The formula used is: \( E = t_{\alpha/2} \times \frac{s}{\sqrt{n}} \).
• We know: \( t_{\alpha/2} = 2.624 \), \( s = 6.53 \) (the sample standard deviation), and \( n = 15 \).
• Plugging these into the formula, we get \( E \approx 4.42 \).

Thus, our margin of error is roughly 4.42 minutes, which expands our estimate to better capture the true mean.

Sample Standard Deviation

The sample standard deviation provides a measure of the spread of the data around the sample mean. It tells us how much individual data points differ from the mean value, effectively indicating the level of variability within a sample.
To find the sample standard deviation:

• Compute each data point's deviation from the mean (e.g., \(x_i - \bar{x}\) where \(\bar{x}\) is the sample mean).
• Square each deviation to avoid negative values.
• Sum all squared deviations.
• Divide by \( n - 1 \) (where \( n \) is the sample size) to get the variance. This is because it's a sample, not a full population, which is why we use \( n - 1 \).
• Take the square root of the variance to get the standard deviation, \( s \approx 6.53 \).

This standard deviation informs us about the diversity of the sample data and is crucial for calculating the margin of error.