Problem 46 - Basic Statistics for Business and Economics with Student

Question

As a condition of employment, Fashion Industries applicants must pass a drug test. Of the last 220 applicants 14 failed the test. Develop a 99 percent confidence interval for the proportion of applicants that fail the test. Would it be reasonable to conclude that more than 10 percent of the applicants are now failing the test? In addition to the testing of applicants, Fashion Industries randomly tests its employees throughout the year. Last year in the 400 random tests conducted, 14 employees failed the test. Would it be reasonable to conclude that less than 5 percent of the employees are not able to pass the random drug test?

Step-by-Step Solution

Answer

It's not reasonable to conclude more than 10% of applicants fail or less than 5% of employees fail.

Step 1: Calculate the Sample Proportion for Applicants

We begin by calculating the sample proportion of applicants who failed the test. Out of 220 applicants, 14 failed, so the sample proportion \( \hat{p} \) is calculated as \( \hat{p} = \frac{14}{220} = 0.0636 \).

Step 2: Calculate the Standard Error for Applicants

Next, we calculate the standard error using the formula: \[ SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \]Substituting the known values, we find:\[ SE = \sqrt{\frac{0.0636 \times (1-0.0636)}{220}} \approx 0.0167 \]

Step 3: Determine the Z-Score for 99% Confidence Interval

For a 99% confidence interval, the Z-score is approximately 2.576.

Step 4: Calculate the Confidence Interval for Applicants

Using the formula for the confidence interval:\[ CI = \hat{p} \pm Z \times SE \]we substitute the values to get:\[ CI = 0.0636 \pm 2.576 \times 0.0167 \]\[ CI = (0.0636 - 0.0430, 0.0636 + 0.0430) \]Thus, the interval is approximately \( (0.0206, 0.1066) \).

Step 5: Analyze if More than 10% Applicants Fail

Check if the confidence interval includes 0.10. Since 0.1066 is just slightly above 0.10, this interval barely accommodates more than 10%, but not conclusively.

Step 6: Calculate the Sample Proportion for Employees

For the employees, the sample proportion \( \hat{p} \) is calculated as \( \hat{p} = \frac{14}{400} = 0.035 \).

Step 7: Calculate the Standard Error for Employees

Calculate the standard error using:\[ SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \]\[ SE = \sqrt{\frac{0.035 \times (1-0.035)}{400}} \approx 0.0092 \]

Step 8: Calculate the Confidence Interval for Employees

Using the confidence interval formula:\[ CI = \hat{p} \pm Z \times SE \] for a 99% confidence interval, we have:\[ CI = 0.035 \pm 2.576 \times 0.0092 \]\[ CI = (0.035 - 0.0237, 0.035 + 0.0237) \]which results in \( (0.0113, 0.0587) \).

Step 9: Analyze if Less than 5% Employees Fail

Check if 0.05 is within the confidence interval. The interval includes 0.05, meaning it would not be reasonable to conclude less than 5% of employees fail based on this 99% confidence.

Key Concepts

Understanding Sample Proportion

When you're looking to understand a group of people, like applicants at Fashion Industries, it's helpful to calculate what we call a sample proportion. Here, we observed 14 out of 220 applicants failing a test. The sample proportion, represented as \( \hat{p} \), is found by dividing the number of individuals meeting the condition (failing the test) by the total number of individuals in the sample (applicants).
For our example, the sample proportion \( \hat{p} \) is \( \frac{14}{220} = 0.0636 \).
This proportion helps us understand how prevalent a trait or behavior is within the sample. It's a crucial first step in analyzing any large group.

Calculating Standard Error

The standard error measures how much sample proportions are expected to fluctuate due to random chance when we take different samples. It's like an estimate of our estimate's 'wiggle room'.

To find the standard error, we use the formula: \[ SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \]
With \( \hat{p} = 0.0636 \) and \( n = 220 \) for the applicants, we find:\[ SE = \sqrt{\frac{0.0636 \times (1-0.0636)}{220}} \approx 0.0167 \]
This tells us how much the sample proportion might differ from one sample to the next, just by chance.

Basics of Hypothesis Testing

Hypothesis testing is a method used to decide if there is enough evidence in our sample to conclude something about the larger group. We started by assuming a hypothesis, which we test using data.

For example, we might want to test if more than 10% of applicants are failing the test. Here, the null hypothesis \( H_0 \) could be that the true proportion is 10%. If our confidence interval doesn't overlap with this 10%, we might reject this null hypothesis, suggesting something else is going on.
This process involves comparing the observed result with what we'd expect if the null hypothesis were true.

Role of the Z-score in Confidence Intervals

A Z-score is a statistical measure that shows how far away a point is from the average. In confidence intervals, it helps us think about how "confident" we can be about the range we've calculated.

For a 99% confidence interval, a Z-score signifies that we expect the true population proportion to lie within \( 2.576 \) standard errors of our sample proportion, on either side. This Z-score comes from a standard normal distribution, helping to quantify our uncertainty.
When we calculate the confidence interval, like for applicants, we use this Z-score to add and subtract from the sample proportion. The formula \[ CI = \hat{p} \pm Z \times SE \]uses \( Z = 2.576 \) to ensure a 99% confidence level, and by plugging values, you create a range where the true proportion likely exists.