Question

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.

Answer 1

Answer:

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

To Find the z  value:

Degree of freedom = n-1

=>81- 1

=> 80

Significance level = 1- confidence level

=>$\frac{(1-\frac{90}{100})}{2}$

=>$\frac{(1-0.90)}{2}$

=> $\frac{0.1}{2}$

=>0.05

using this value In T- Distribution table we get

z =  1.645

Substituting the values  we have,

confidence interval = $3408\pm 1.645 * \frac{2565}{\sqrt{81}}$

confidence interval = $3408\pm 1.645 * \frac{2565}{\sqrt{9}}$

confidence interval = $3408\pm 1.645 * 285$

confidence interval = $3408\pm 468.825$

confidence interval= (2939.18, 3876.83)

Answer 2

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 https://brainly.com/question/15180581