Problem 15 - Basic Statistics for Business & Economics

Question

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

Step-by-Step Solution

Answer

a. Population proportion estimate: 0.8. b. 95% confidence interval: (0.7216, 0.8784). c. Interpretation: 95% confidence the true proportion is within the interval.

Step 1: Identify Given Information

We have a sample size of 100 customers and 80 of them pay at the pump using a card. Let the sample size be denoted by \(n = 100\) and the number of successes (paying with a card) be \(x = 80\).

Step 2: Estimate the Population Proportion

The sample proportion \(\hat{p}\) is calculated as \(\hat{p} = \frac{x}{n} = \frac{80}{100} = 0.8\). Therefore, the estimated population proportion is 0.8.

Step 3: Calculate the Standard Error

The standard error (SE) of the sample proportion can be calculated using the formula \(SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} = \sqrt{\frac{0.8 \times 0.2}{100}} = 0.04\).

Step 4: Determine the Critical Value for a 95% Confidence Interval

For a 95% confidence interval, the critical value \(z\) is approximately 1.96. This value is derived from the standard normal distribution.

Step 5: Calculate the Confidence Interval

The confidence interval for the population proportion is given by \(\hat{p} \pm z \times SE\). Calculating this gives us \(0.8 \pm 1.96 \times 0.04\), which results in a confidence interval of \( (0.7216, 0.8784) \).

Step 6: Interpret the Confidence Interval

The 95% confidence interval suggests that we are 95% confident that the true proportion of customers who pay at the pump with a card is between 72.16% and 87.84%.

Key Concepts

Population Proportion

When an owner wants to understand the behavior of their customers, looking at the population proportion gives a good start. The population proportion represents the fraction of total elements in the population that exhibit a certain characteristic. In the gas station scenario, it answers the question: "What proportion of all customers pay at the pump using a credit or debit card?"
The owner can't survey every customer due to practical limitations, so they use a **sample proportion** as an estimate. This involves dividing the number of customers observed (in this case 80) by the total number of surveyed customers (100). Thus, the estimated population proportion is 0.8 or 80%.

• Population proportion reflects the general habit of the whole customer base, while sample proportion gives a snapshot through a specific group.

Sample Size

Sample size, denoted as **n**, is crucial in surveys for its impact on the accuracy of estimating the population proportion. Here, the owner surveyed 100 customers at the gas station. A larger sample size generally leads to a more accurate estimate of the population proportion because it reduces sampling error.
When considering sample size:

• It directly influences the reliability of the estimate; a larger n offers a closer approximation to the true population characteristics.
• The law of large numbers explains that increasing the sample size will usually provide more stable and accurate estimates.

This makes an adequately large sample important for meaningful insights and reduces the likelihood of misleading results from random variances.

Standard Error

Standard error (SE) is a measure of the variation or "spread" of sample proportion estimates around the true population proportion. It helps quantify how much the sample proportion (0.8 in this case) might differ from the actual population proportion.
To calculate the standard error, use the formula: \[ SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \] where \( \hat{p} \) is the sample proportion and \( n \) is the sample size.
For our example, this calculation leads to 0.04. A smaller standard error indicates that the sample proportion is likely closer to the population proportion, providing more precise estimates. Understanding this allows the owner to gauge the reliability of their findings.

Critical Value

In statistics, the critical value helps determine the range (confidence interval) within which the true population proportion likely lies. For a **95% confidence level**, the critical value is typically 1.96, derived from the standard normal distribution.
This value means that if you were to take 100 different samples and compute a confidence interval from each, 95 of those intervals would contain the true population proportion.
When computing the interval in our exercise, the critical value shapes the width of the interval:\[ \hat{p} \pm z \times SE \]Here, \( z = 1.96 \) and SE is 0.04, leading to an interval of (0.7216, 0.8784).
Recognizing the role of the critical value enables an understanding of how confident you can be in your interpretation of the sample data, using probabilities to ensure error margins remain within acceptable bounds.