Math / Data & Statistics

QuestionIn a random sample of 49 audited estate tax returns, it was determined that the mean amount of additional tax owed was $\$ 3421$ with a standard deviation of $\$ 2582$ . Construct and interpret a $90 \%$ confidence interval for the mean additional amount of tax owed for estate tax returns.
Find and interpret a 90\% confidence interval for the mean additional amount of tax owed for estate tax returns. Select the correct choice below and fill in the answer boxes to complete your choice. (Use ascending order. Round to the nearest dollar as needed.) A. There is a $90 \%$ probability that the mean additional tax owed is between $\$$ $\square$ and \\square $B. One can be$ 90 \% $confident that the mean additional tax owed is between$ \$$ $\square$ and \$ C. $90 \%$ of taxes owed for estate tax returns are between $\$$ $\square$ and \$ $\square$.

Studdy Solution

STEP 1

1. The sample size is 49, which is sufficiently large for the Central Limit Theorem to apply.
2. The sample mean of additional tax owed is \$3421.
3. The sample standard deviation is \$2582.
4. We are constructing a 90% confidence interval for the population mean.

STEP 2

1. Identify the critical value for a 90% confidence interval.
2. Calculate the standard error of the mean.
3. Compute the confidence interval.
4. Interpret the confidence interval.

STEP 3

Identify the critical value (z-score) for a 90% confidence interval. For a 90% confidence level, the critical value (z) is approximately 1.645.

STEP 4

Calculate the standard error of the mean (SEM) using the formula:
$\text{SEM} = \frac{\text{standard deviation}}{\sqrt{\text{sample size}}} = \frac{2582}{\sqrt{49}} = \frac{2582}{7} \approx 369.14$

STEP 5

Compute the confidence interval using the formula:
$\text{Confidence Interval} = \text{mean} \pm (\text{critical value} \times \text{SEM})$
$= 3421 \pm (1.645 \times 369.14)$
Calculate the margin of error:
$1.645 \times 369.14 \approx 607.43$
Calculate the confidence interval:
$3421 - 607.43 \approx 2813.57$
$3421 + 607.43 \approx 4028.43$
Round to the nearest dollar:
$\text{Confidence Interval} = (2814, 4028)$

STEP 6

Interpret the confidence interval. The correct choice is:
B. One can be $90\%$ confident that the mean additional tax owed is between $\$2814$ and $\$4028$ .
The 90% confidence interval for the mean additional amount of tax owed is $\$2814$ to $\$4028$ .

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Studdy Solution

STEP 1

1. The sample size is 49, which is sufficiently large for the Central Limit Theorem to apply.
2. The sample mean of additional tax owed is \$3421.
3. The sample standard deviation is \$2582.
4. We are constructing a 90% confidence interval for the population mean.

STEP 2

1. Identify the critical value for a 90% confidence interval.
2. Calculate the standard error of the mean.
3. Compute the confidence interval.
4. Interpret the confidence interval.

STEP 3

Identify the critical value (z-score) for a 90% confidence interval. For a 90% confidence level, the critical value (z) is approximately 1.645.

STEP 4

Calculate the standard error of the mean (SEM) using the formula:
$\text{SEM} = \frac{\text{standard deviation}}{\sqrt{\text{sample size}}} = \frac{2582}{\sqrt{49}} = \frac{2582}{7} \approx 369.14$

STEP 5

STEP 6

Interpret the confidence interval. The correct choice is:
B. One can be $90\%$ confident that the mean additional tax owed is between $\$2814$ and $\$4028$ .
The 90% confidence interval for the mean additional amount of tax owed is $\$2814$ to $\$4028$ .

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Studdy Solution

STEP 1

1. The sample size is 49, which is sufficiently large for the Central Limit Theorem to apply.2. The sample mean of additional tax owed is \$3421.3. The sample standard deviation is \$2582.4. We are constructing a 90% confidence interval for the population mean.

STEP 2

1. Identify the critical value for a 90% confidence interval.2. Calculate the standard error of the mean.3. Compute the confidence interval.4. Interpret the confidence interval.

STEP 3

Identify the critical value (z-score) for a 90% confidence interval. For a 90% confidence level, the critical value (z) is approximately 1.645.

STEP 4

STEP 5

STEP 6

Interpret the confidence interval. The correct choice is:B. One can be 90%90\%90% confident that the mean additional tax owed is between $2814\$2814$2814 and $4028\$4028$4028. The 90% confidence interval for the mean additional amount of tax owed is $2814\$2814$2814 to $4028\$4028$4028.

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Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. The sample size is 49, which is sufficiently large for the Central Limit Theorem to apply.
2. The sample mean of additional tax owed is \$3421.
3. The sample standard deviation is \$2582.
4. We are constructing a 90% confidence interval for the population mean.

1. Identify the critical value for a 90% confidence interval.
2. Calculate the standard error of the mean.
3. Compute the confidence interval.
4. Interpret the confidence interval.

Interpret the confidence interval. The correct choice is:
B. One can be $90\%$ confident that the mean additional tax owed is between $\$2814$ and $\$4028$ .
The 90% confidence interval for the mean additional amount of tax owed is $\$2814$ to $\$4028$ .