Mathematics, 26.11.2019 19:31 ShilohTheBoy

In a random sample of 81 audited estate tax returns, it was determined that the mean amount of additional tax owed was $3408 with a standard deviation of $2565. construct and interpret a 90% confidence interval for the mean additional amount of tax owed for estate tax returns. the lower bound is $ nothing. (round to the nearest cent as needed.) the upper bound is $ nothing. (round to the nearest cent as needed.) interpret a 90% confidence interval for the mean additional amount of tax owed for estate tax returns. choose the correct answer below.

a. one can be 90% confident that the mean additional tax owed is less than the lower bound
b. one can be 90% confident that the mean additional tax owed is between the lower and upper bounds
c. one can be 90% confident that the mean additional tax owed is greater than the upper bound.

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Answer from: bryce860

Using the t-distribution, it is found that:

The lower bound is $2,933.73.The upper bound is $3,822.27.

b. One can be 90% confident that the mean additional tax owed is between the lower and upper bounds

We are given the standard deviation for the sample, which is why the t-distribution is used to solve this question.

The information given is:

Sample mean of $\overline{x} = 3408$ .

Sample standard deviation of s = 2565 .

Sample size of n = 81 .

The confidence interval is:

$\overline{x} \pm t\frac{s}{\sqrt{n}}$

The critical value, using a t-distribution calculator, for a two-tailed 90% confidence interval, with 81 - 1 = 80 df, is t = 1.6641.

Then, the bounds of the interval are:

$\overline{x} - t\frac{s}{\sqrt{n}} = 3408 - 1.6641\frac{2565}{\sqrt{81}} = 2933.73$

$\overline{x} + t\frac{s}{\sqrt{n}} = 3408 + 1.6641\frac{2565}{\sqrt{81}} = 3822.27$

The lower bound is $2,933.73.

The upper bound is $3,822.27.

The interpretation is that we are 90% sure that the true mean is in the interval, that is, option b.

A similar problem is given at link

Answer from: kat2788

B. One can be 90% confident that the mean additional tax owed is between the lower and upper bounds.

Step-by-step explanation:

Given:

n= 81

$\bar{x}=3408$

$\sigma= 2565$

Solution:

A confidence interval, in statistics, refers to the probability that a population parameter will fall between two set values for a certain proportion of times. Confidence intervals measure the degree of uncertainty or certainty in a sampling method. A confidence interval can take any number of probabilities, with the most common being a 95% or 99% confidence level.

Confidence interval = $\bar{x} \pm z * \frac{\sigma}{\sqrt{n}}$