Question

In a random sample of 100 audited estate tax​ returns, it was determined that the mean amount of additional tax owed was $3444 with a standard deviation of $2504.

Construct and interpret a​ 90% confidence interval for the mean additional amount of tax owed for estate tax returns.

The lower bound is _____. (Round to the nearest dollar as​needed.)

The upper bound is ______. (Round to the nearest dollar as​needed.)

Answer 1

The lower bound of the 90% confidence interval for the mean additional amount of tax owed for estate tax returns is approximately $3141. The upper bound is approximately $3747.

Step-by-step explanation:

To construct a confidence interval for the mean, we use the formula:
`\(\bar{X} \pm Z \cdot (\sigma)/(√(n))\)`, where
`\(\bar{X}\)` is the sample mean,
`\(\sigma\)` is the population standard deviation,
`\(n\)` is the sample size, and
`\(Z\)` is the Z-score corresponding to the desired confidence level. For a 90% confidence interval, the Z-score is approximately 1.645.

Given a sample mean
`(\(\bar{X}\))` of $3444, a standard deviation
`(\(\sigma\))` of $2504, and a sample size
`(\(n\))` of 100, we can substitute these values into the formula. Calculating the margin of error and then determining the lower and upper bounds of the confidence interval, we get approximately $3141 and $3747, respectively.

Interpreting the confidence interval, we can say with 90% confidence that the true mean additional amount of tax owed for estate tax returns falls between $3141 and $3747. This means that if we were to take many random samples and calculate the confidence intervals, about 90% of them would contain the true population mean. The bounds provide a range of values within which we estimate the actual mean to lie.