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.

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 = [tex]\bar{x} \pm z * \frac{\sigma}{\sqrt{n}}[/tex]

To Find the z value:

Degree of freedom = n-1

=>81- 1

=> 80

Significance level = 1- confidence level

=>[tex]\frac{(1-\frac{90}{100})}{2}[/tex]

=>[tex]\frac{(1-0.90)}{2}[/tex]

=> [tex]\frac{0.1}{2}[/tex]

=>0.05

using this value In T- Distribution table we get

z = 1.645

Substituting the values we have,

confidence interval = [tex]3408\pm 1.645 * \frac{2565}{\sqrt{81}}[/tex]

confidence interval = [tex]3408\pm 1.645 * \frac{2565}{\sqrt{9}}[/tex]

confidence interval = [tex]3408\pm 1.645 * 285[/tex]

confidence interval = [tex]3408\pm 468.825[/tex]

confidence interval= (2939.18, 3876.83)

joaobezerra joaobezerra · Answer 2 · 2021-11-11T13:16:36+01:00

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 [tex]\overline{x} = 3408[/tex].

Sample standard deviation of [tex]s = 2565[/tex].

Sample size of [tex]n = 81[/tex].

The confidence interval is:

[tex]\overline{x} \pm t\frac{s}{\sqrt{n}}[/tex]

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:

[tex]\overline{x} - t\frac{s}{\sqrt{n}} = 3408 - 1.6641\frac{2565}{\sqrt{81}} = 2933.73[/tex]

[tex]\overline{x} + t\frac{s}{\sqrt{n}} = 3408 + 1.6641\frac{2565}{\sqrt{81}} = 3822.27[/tex]

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 https://brainly.com/question/15180581