Problem 15 - Basic Statistics for Business and Economics with Student

Question

15\. The owner of the West End Kwick Fill Gas Station wishes to determine the proportion of customers who use a credit card or debit card to pay at the pump. He surveys 100 customers and finds that 80 paid with a credit card or a debit card at the pump. a. Estimate the value of the population proportion. b. Develop a 95 percent confidence interval for the population proportion. c. Interpret your findings.

Step-by-Step Solution

Answer

Estimated proportion is 0.80; 95% CI is (0.7216, 0.8784).

Step 1: Identify the Sample Proportion

The sample proportion (\( \hat{p} \)) is the ratio of the number of customers who paid using a credit or debit card to the total number of surveyed customers. Since 80 out of 100 customers paid with a card, \( \hat{p} = \frac{80}{100} = 0.80 \).

Step 2: Formula for the Confidence Interval

To find a 95% confidence interval for the population proportion, we use:\[\hat{p} \pm Z \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\]where \( Z \) is the Z-value for the desired confidence level (1.96 for 95%).

Step 3: Calculate the Standard Error

Calculate the standard error using \( \hat{p} = 0.80 \) and \( n = 100 \):\[SE = \sqrt{\frac{0.80 \times (1 - 0.80)}{100}} = \sqrt{\frac{0.80 \times 0.20}{100}} = \sqrt{\frac{0.16}{100}} = 0.04\]

Step 4: Margin of Error Calculation

Calculate the margin of error using the standard error:\[MOE = Z \times SE = 1.96 \times 0.04 = 0.0784\]

Step 5: Construct the Confidence Interval

With \( \hat{p} = 0.80 \) and \( MOE = 0.0784 \), the confidence interval is:\[(0.80 - 0.0784, 0.80 + 0.0784) = (0.7216, 0.8784)\]

Step 6: Interpretation of the Confidence Interval

We are 95% confident that the true population proportion of customers using a credit or debit card at the pump is between 0.7216 and 0.8784.

Key Concepts

Confidence Interval

A confidence interval is like a safety net for our estimates. It tells us how confident we can be about the range where the true population parameter lies. For example, when a gas station owner wants to know what proportion of customers will use a card at the pump, he doesn't have data for every single customer. Instead, he surveys 100 customers. Since it would be nearly impossible to survey everyone, a confidence interval helps us make an informed guess about the population's behavior with the limited data we have.

The confidence interval is calculated using the sample proportion and a Z-value, which depends on how confident we want to be (in this case, 95%). The formula is:

• \[ \hat{p} \pm Z \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \ \]

Here, \( \hat{p} \) is the sample proportion, \( Z \) is the Z-value, and \( n \) is the total number of surveyed customers. For our case, it indicates the range (0.7216, 0.8784) within which we are 95% confident that the true population proportion lies.

This means we can be quite sure (though not absolutely certain) that the true proportion of customers paying by card is somewhere between those numbers.

Sample Proportion

The sample proportion (\(\hat{p}\) is a snapshot of the population based on the survey. It tells us the likelihood of an event happening in your sample group. Think of it as the percentage of customers using a card to pay at the pump.

In our gas station example, the owner found out that out of 100 customers, 80 used cards. Therefore, the sample proportion is calculated as:

• \[\hat{p} = \frac{80}{100} = 0.80 \]

This means 80% of the surveyed customers paid with a card. It gives a quick and handy way to estimate how many would probably use a card if we surveyed a larger group.

However, remember that the sample proportion is based on the sampled group and not the entire population. Hence, it may differ slightly when different samples are considered.

Margin of Error

The margin of error acts as a cushion around our estimate. It measures how much we can expect the sample proportion to vary from the true population proportion. This helps us understand the potential error range in our estimate so that we don't overstate our claims.

To calculate the margin of error, we need the standard error and Z-value. Using our surveyed data, we found the standard error was 0.04. Multiplying it by the Z-value (1.96 for 95% confidence) gives:

• Margin of Error = \[ Z \times SE = 1.96 \times 0.04 = 0.0784 \ \]

This means, based on our sample, we can expect the actual proportion of card-using customers to fall within 7.84 percentage points of our sample proportion.

The smaller the margin of error, the more precise our confidence interval becomes. It’s essential in statistical studies since it provides a clearer picture of how reliable our estimated result is likely to be.