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Magazine Chapter 9: Problem 50
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.
Short Answer
Expert verified
The 99% confidence interval for the proportion is approximately (0.34, 0.70).
Step by step solution
01
Define the problem
We are asked to find a 99% confidence interval for the proportion of checking account customers at the First National Bank of Wilson who have a Visa card with the bank.
02
Determine sample proportion
The sample proportion \( \hat{p} \) is calculated using the formula \( \hat{p} = \frac{x}{n} \), where \( x \) is the number of successes (customers with a Visa card) and \( n \) is the sample size. Here, \( x = 26 \) and \( n = 50 \), so \( \hat{p} = \frac{26}{50} = 0.52 \).
03
Find the critical value
Since we are calculating a 99% confidence interval, we need the critical value \( Z \) associated with this confidence level. The critical value for a 99% confidence level from the standard normal distribution is approximately \( Z = 2.576 \).
04
Calculate the standard error
The standard error \( SE \) for the proportion is calculated using the formula \( SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \). In this case, \( SE = \sqrt{\frac{0.52(1 - 0.52)}{50}} = \sqrt{\frac{0.4992}{50}} \approx 0.070 \).
05
Construct the confidence interval
The 99% confidence interval is given by \( \hat{p} \pm Z \times SE \). Therefore, the interval is \( 0.52 \pm 2.576 \times 0.070 \), which simplifies to \(0.52 \pm 0.18032\). The interval is (0.33968, 0.70032).
06
Interpret the results
We can be 99% confident that the true proportion of checking account customers at the First National Bank of Wilson who have a Visa card is between approximately 0.34 and 0.70.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Proportion
In statistics, the term 'proportion' refers to the fraction or percentage of a specific group or category within a larger population. For instance, when determining the proportion of checking account customers who own a Visa card in a sample, we divide the number of customers with Visa cards by the total number of customers surveyed.
In the context of our exercise, 26 customers out of the 50 surveyed have a Visa card. This proportion is calculated as follows:
In the context of our exercise, 26 customers out of the 50 surveyed have a Visa card. This proportion is calculated as follows:
- \[ \hat{p} = \frac{26}{50} = 0.52 \]
This means that 52% of the sample have a Visa card with the bank. Understanding proportions allows researchers to make inferences about the population based on the sample data. - Proportions are essential for calculating confidence intervals and identifying trends and habits in customer behavior.
The concept is foundational for the computations that follow, such as standard error and confidence intervals.
Standard Error
The Standard Error (SE) quantifies the uncertainty of the sample proportion. It helps in understanding how much the proportion found in the sample may deviate from the actual proportion in the overall population.
The formula for the standard error of a sample proportion is:
The formula for the standard error of a sample proportion is:
- \[ SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \]
- \[ SE = \sqrt{\frac{0.52 \times (1 - 0.52)}{50}} = \sqrt{\frac{0.2496}{50}} \approx 0.070 \]
Here, \( \hat{p} \) is the sample proportion (0.52 in our example), and \( n \) is the sample size (50).
In our problem, substituting the values:
As the standard error helps in constructing the confidence interval, knowing this value indicates the variability you might expect if you were to take another sample of the same size. A smaller standard error suggests less variability and thus more precision in estimating the population proportion.
Critical Value
The critical value in statistical analysis helps us define the range of our confidence interval. It indicates the number of standard errors to extend from the calculated sample proportion to cover a certain confidence level.
For our 99% confidence interval, the critical value is associated with how confident we want to be about our interval. With higher confidence levels, the critical value increases.
For a 99% confidence level, the critical value \( Z \) from the standard normal distribution is approximately 2.576.
This means that, in our calculations:
For our 99% confidence interval, the critical value is associated with how confident we want to be about our interval. With higher confidence levels, the critical value increases.
For a 99% confidence level, the critical value \( Z \) from the standard normal distribution is approximately 2.576.
This means that, in our calculations:
- \( \hat{p} \pm Z \times SE \)
- \( 0.52 \pm 2.576 \times 0.070 \)
- \( 0.52 \pm 0.18032 \)
Therefore, the critical value determines the width of the confidence interval, and thus how accurately we feel our sample proportion predicts the population proportion.
Sample Proportion
The Sample Proportion, denoted as \( \hat{p} \), is an estimate of the true proportion of the population derived from a sample. It is essential for making statistical estimates about the population as a whole.
In the exercise, our sample proportion tells us what fraction of our sample (of checking account customers) owns a Visa card with the bank. Here is how it is calculated:
In the exercise, our sample proportion tells us what fraction of our sample (of checking account customers) owns a Visa card with the bank. Here is how it is calculated:
- \( \hat{p} = \frac{x}{n} \)
- \( \hat{p} = \frac{26}{50} = 0.52 \)
Where \( x \) is the number of customers with Visa cards (26), and \( n \) is the sample size (50).
So in this context:
This indicates that, based on the sample, approximately 52% of the bank's checking account customers may have a Visa card. The sample proportion is critical because it is the backbone of further calculations, such as the confidence interval, which generalizes this sample proportion to the whole population.
