Problem 22 - Basic Statistics for Business and Economics with Student

Question

There are 300 welders employed at Maine Shipyards Corporation. A sample of 30 welders revealed that 18 graduated from a registered welding course. Construct the 95 percent confidence interval for the proportion of all welders who graduated from a registered welding course.

Step-by-Step Solution

Answer

The 95% confidence interval for the proportion is approximately [0.4248, 0.7752].

Step 1: Determine the Sample Proportion

First, calculate the sample proportion \( \hat{p} \) of welders who graduated from a registered welding course. Given that, 18 out of 30 welders graduated, the sample proportion is \( \hat{p} = \frac{18}{30} = 0.6 \).

Step 2: Find the Z-Score for the Confidence Level

The confidence level given is 95 percent. For a 95% confidence interval, the Z-score corresponding to this confidence level is 1.96.

Step 3: Calculate the Standard Error

The standard error (SE) of the sample proportion is calculated using the formula \( SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \), where \( n \) is the sample size. Substitute \( \hat{p} = 0.6 \) and \( n = 30 \):\[SE = \sqrt{\frac{0.6(1-0.6)}{30}} = \sqrt{\frac{0.6 \times 0.4}{30}} = \sqrt{\frac{0.24}{30}} = \sqrt{0.008} \approx 0.0894\]

Step 4: Determine the Margin of Error

Calculate the margin of error (ME) using the formula \( ME = Z \times SE \). Using the Z-score from Step 2 and the standard error from Step 3:\[ME = 1.96 \times 0.0894 \approx 0.1752\]

Step 5: Calculate the Confidence Interval

The 95% confidence interval is found using the formula \( \hat{p} \pm ME \). Thus, \[CI = 0.6 \pm 0.1752\]This results in the interval \( [0.4248, 0.7752] \). This means we are 95% confident that the true proportion of welders who graduated lies between 0.4248 and 0.7752.

Key Concepts

Sample Proportion

When we talk about the sample proportion, we're looking at a simple way to understand what part of a group has a particular characteristic. In our exercise, the group is the welders at Maine Shipyards, and the characteristic is graduating from a registered welding course. To find the sample proportion, we take the number of successes (welders who graduated, which is 18) and divide it by the total number of observations (the sample size, which is 30). So, the sample proportion is calculated as:

• \( \hat{p} = \frac{18}{30} = 0.6 \)

This means that in our sample, 60% of the welders have graduated from the course.This proportion helps us estimate how common graduation might be in the whole group of 300 welders by using their sample as a basic reflection of the larger population.

Standard Error

The standard error measures how much we can expect the sample proportion to vary from the true population proportion. It gives us an idea of the estimate's precision from the sample data.To calculate the standard error of the sample proportion, use the formula:

• \( SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \)

In our example, \( \hat{p} \) is 0.6, and \( n \) is 30.Plug these values into the formula:\[SE = \sqrt{\frac{0.6 \times 0.4}{30}} = \sqrt{\frac{0.24}{30}} = \sqrt{0.008} \approx 0.0894\]The standard error is approximately 0.0894.
This number tells us about the likelihood of variation in the sample proportion.
A smaller standard error means less dispersion around the true proportion.

Margin of Error

The margin of error gives us the range around the sample proportion within which we can expect the true population proportion to lie. It is directly linked to the standard error and the Z-score of the desired confidence level.To find the margin of error, multiply the standard error by the Z-score:

• \( ME = Z \times SE \)

In this case, the Z-score for a 95% confidence level is 1.96.Using our previous standard error of 0.0894:\[ME = 1.96 \times 0.0894 \approx 0.1752\]The margin of error is approximately 0.1752. So, our sample proportion could swing 17.52% either side of what we've measured.
This helps us quantify the confidence we have in the sample proportion as an estimator for the population proportion.

Z-score

The Z-score is a statistical measure that describes how many standard deviations a data point is from the mean. In a confidence interval context, it helps us determine the range within which we can expect the true population parameter to lie. For a 95% confidence interval, the commonly used Z-score is 1.96. This number corresponds to a significance level where only 2.5% of values lie beyond this mark on either side of a normal distribution curve. The Z-score helps in determining the confidence level:

• A higher Z-score gives more certainty, but a wider interval.
• A lower Z-score gives less certainty, but a tighter interval.

By combining this Z-score with the standard error, we can calculate the margin of error, which in turn helps us build a confidence interval around the sample estimate.