Problem 50 - Basic Statistics for Business and Economics with Student

Question

The First National Bank of Wilson has 650 checking account customers. A recent sample of 50 of these customers showed 26 to have a Visa card with the bank. Construct the 99 percent confidence interval for the proportion of checking account customers who have a Visa card with the bank.

Step-by-Step Solution

Answer

The 99% confidence interval is approximately [0.34, 0.70].

Step 1: Identify Sample Proportion

First, calculate the sample proportion of customers with a Visa card. This is given by the formula \( \hat{p} = \frac{x}{n} \), where \( x \) is the number of successes (customers with Visa cards) and \( n \) is the sample size. For this problem, \( x = 26 \) and \( n = 50 \). Therefore, the sample proportion \( \hat{p} = \frac{26}{50} = 0.52 \).

Step 2: Determine the Z-Score for Confidence Level

For a 99% confidence interval, you need to find the Z-score that corresponds to a 99% confidence level. From statistical tables, the Z-score for 99% confidence level is approximately 2.576.

Step 3: Calculate the Standard Error

The standard error (SE) of the sample proportion is calculated using the formula \( SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \). Substituting the values, we get \( SE = \sqrt{\frac{0.52 \times (1 - 0.52)}{50}} \approx 0.070 \).

Step 4: Compute the Confidence Interval

Now, calculate the confidence interval using the formula \( \hat{p} \pm Z \times SE \). Substituting the values gives \( 0.52 \pm 2.576 \times 0.070 \). This results in an interval of approximately \( [0.34, 0.70] \).

Step 5: Interpret the Results

The 99% confidence interval for the proportion of all checking account customers who have a Visa card with the bank is approximately \([0.34, 0.70]\). This means we can be 99% confident that the true proportion of all checking account customers with a Visa card is between 34% and 70%.

Key Concepts

Understanding Sample Proportion

Sample proportion is a key concept in statistics used to make inferences about a population characteristic based on a sample. In this exercise, the sample proportion, denoted as \( \hat{p} \), represents the fraction of the sample that holds a certain attribute—in this case, the customers holding a Visa card from the bank.
Computing the sample proportion is straightforward. You use the formula \( \hat{p} = \frac{x}{n} \), where \( x \) is the number of events of interest (Visa cardholders) and \( n \) is the total number of observations (total sample size).
For example:

• The bank had 50 customers in the sample.
• 26 held a Visa card.
• The sample proportion (\(\hat{p}\)) is calculated as \( \frac{26}{50} = 0.52 \).

This means that 52% of the sampled customers have a Visa card from the bank. Understanding the sample proportion helps in estimating the population proportion, which is ultimately what the confidence interval will provide.

Role of Standard Error in Confidence Intervals

The standard error (SE) measures the variability or dispersion of the sample proportion from the true population proportion. It is crucial for determining how accurately the sample proportion estimates the population proportion.
The standard error for the sample proportion is computed using the formula \( SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \), where \( \hat{p} \) is the sample proportion and \( n \) is the sample size.
Using the example:

• \( \hat{p} = 0.52 \)
• \( n = 50 \)
• The standard error is \( SE = \sqrt{\frac{0.52 \times (1-0.52)}{50}} \approx 0.070 \).

The smaller the standard error, the more precise the estimate of the population proportion. This precision forms a critical part of the confidence interval calculation, allowing us to quantify the uncertainty associated with the sample estimate.

Explaining Z-score for 99% Confidence

The Z-score is a statistical measure that reflects the number of standard deviations a data point is from the mean, in the context of the standard normal distribution. For confidence intervals, the Z-score helps determine the range of values the true parameter lies within, given a level of confidence.
For a 99% confidence interval, a Z-score of approximately 2.576 is used. This number is derived from statistical Z-tables and indicates that we expect the true population parameter to fall within 2.576 standard deviations of the sample mean.
When you apply this Z-score to our example, you multiply it by the standard error to find the margin of error for the interval. The margin of error is then used to adjust our sample proportion to create the confidence interval:

• Margin of error = \( Z \times SE = 2.576 \times 0.070 \)

The resulting interval around the sample proportion tells us where the true population proportion is likely to fall, with our stated confidence.

Interpretation of Results in the Context of Confidence Intervals

Interpreting confidence intervals involves understanding the range and the level of confidence we have that this range captures the true population parameter. In this example, the 99% confidence interval is \([0.34, 0.70]\).
This interval suggests that we can be 99% confident that the actual proportion of all checking account holders who hold a Visa card with the bank will fall anywhere between 34% and 70%.
This doesn't say that 99% of customers have a Visa card, nor does it refer to the likelihood of a single customer having one. Instead, it communicates a range in which we expect the actual population proportion to lie, considering variability in our sample.
Confidence intervals provide a way to express statistical uncertainty and make informed guesses about population parameters based on sample data. By understanding the width of the interval, the sample size, and the chosen confidence level, one can make statistically sound conclusions about the underlying population.