Question

The amounts of debt people have accumulated one year after having started to use their first credit card is normally distributed. Visa takes a sample of 36 people who opened the account one year ago and finds that for this group of people the average amount of debt accumulated was $700. The standard deviation is $180. Find the margin of error for the 95% confidence interval for the mean amount of debt Visa can expect credit card holders to accumulate one year after opening their first account​

Answer 1

Answer: To find the margin of error for a 95% confidence interval for the mean amount of debt Visa can expect credit card holders to accumulate one year after opening their first account, we can use the formula:

Margin of Error = Z * (Standard Deviation / sqrt(n))

Here, Z represents the Z-score, which corresponds to the desired confidence level. For a 95% confidence level, the Z-score is approximately 1.96.

Given information:

- Average amount of debt accumulated: $700

- Standard deviation: $180

- Sample size: 36

Now, let's calculate the margin of error step-by-step:

1. Calculate the Z-score:

Z = 1.96

2. Calculate the standard error:

Standard Error = Standard Deviation / sqrt(n)

Standard Error = $180 / sqrt(36)

Standard Error = $180 / 6

Standard Error = $30

3. Calculate the margin of error:

Margin of Error = Z * Standard Error

Margin of Error = 1.96 * $30

Margin of Error ≈ $58.80

Therefore, the margin of error for the 95% confidence interval for the mean amount of debt Visa can expect credit card holders to accumulate one year after opening their first account is approximately $58.80.

This means that we can be 95% confident that the true mean amount of debt for all credit card holders one year after opening their first account falls within the range of $700 ± $58.80.