      I.        Confidence interval:

       II.            Standard error (SE):

     III.            Z-values:

**Confidence level**

**z-value**

**90%**

1.64

**95%**

1.96

**99%**

2.58
 
1.       We measured the heights of 100 Guyanese men. The mean height of men in the sample is 71 inches, with a standard deviation of 2.8 inches. Construct a

a.       90% confidence interval for the height of Guyanese men.

b.       95% confidence interval for the height of Guyanese men.

c.       99% confidence interval for the height of Guyanese men.

2.       A study was done to measure the time taken to complete tax forms. 100 people were surveyed, and a mean time of 23.6 hours was found to be needed to complete tax forms, with a standard deviation of 7.0 hours. Assume the underlying population is normally distributed. Calculate:

a.       A 90% confidence interval for the time taken to complete tax forms

b.       A 95% confidence interval for the time taken to complete tax forms

c.       A 99% confidence interval for the time taken to complete tax forms

To solve these problems, we need to construct confidence intervals based on the given data. The formula for constructing a confidence interval is:

\[
CI = \bar{X} \pm Z \times \frac{S}{\sqrt{n}}
\]

Where:
- \( \bar{X} \) is the sample mean,
- \( Z \) is the z-value associated with the confidence level,
- \( S \) is the sample standard deviation,
- \( n \) is the sample size,
- \( \frac{S}{\sqrt{n}} \) is the standard error (SE).

Let's solve each part of the question.

### Problem 1: Heights of Guyanese Men

Given:
- Sample size \( n = 100 \),
- Sample mean \( \bar{X} = 71 \) inches,
- Sample standard deviation \( S = 2.8 \) inches.

#### a. 90% Confidence Interval

For 90% confidence, \( Z = 1.64 \).

The standard error \( SE \) is:
\[
SE = \frac{S}{\sqrt{n}} = \frac{2.8}{\sqrt{100}} = 0.28
\]

Now, compute the confidence interval:
\[
CI = 71 \pm 1.64 \times 0.28
\]
\[
CI = 71 \pm 0.4592
\]
\[
CI = (70.5408, 71.4592)
\]

So, the **90% confidence interval** is approximately \( (70.54, 71.46) \).

#### b. 95% Confidence Interval

For 95% confidence, \( Z = 1.96 \).

Now, compute the confidence interval:
\[
CI = 71 \pm 1.96 \times 0.28
\]
\[
CI = 71 \pm 0.5488
\]
\[
CI = (70.4512, 71.5488)
\]

So, the **95% confidence interval** is approximately \( (70.45, 71.55) \).

#### c. 99% Confidence Interval

For 99% confidence, \( Z = 2.58 \).

Now, compute the confidence interval:
\[
CI = 71 \pm 2.58 \times 0.28
\]
\[
CI = 71 \pm 0.7224
\]
\[
CI = (70.2776, 71.7224)
\]

So, the **99% confidence interval** is approximately \( (70.28, 71.72) \).

---

### Problem 2: Time to Complete Tax Forms

Given:
- Sample size \( n = 100 \),
- Sample mean \( \bar{X} = 23.6 \) hours,
- Sample standard deviation \( S = 7.0 \) hours.

#### a. 90% Confidence Interval

For 90% confidence, \( Z = 1.64 \).

The standard error \( SE \) is:
\[
SE = \frac{S}{\sqrt{n}} = \frac{7.0}{\sqrt{100}} = 0.7
\]

Now, compute the confidence interval:
\[
CI = 23.6 \pm 1.64 \times 0.7
\]
\[
CI = 23.6 \pm 1.148
\]
\[
CI = (22.452, 24.748)
\]

So, the **90% confidence interval** is approximately \( (22.45, 24.75) \).

#### b. 95% Confidence Interval

For 95% confidence, \( Z = 1.96 \).

Now, compute the confidence interval:
\[
CI = 23.6 \pm 1.96 \times 0.7
\]
\[
CI = 23.6 \pm 1.372
\]
\[
CI = (22.228, 24.972)
\]

So, the **95% confidence interval** is approximately \( (22.23, 24.97) \).

#### c. 99% Confidence Interval

For 99% confidence, \( Z = 2.58 \).

Now, compute the confidence interval:
\[
CI = 23.6 \pm 2.58 \times 0.7
\]
\[
CI = 23.6 \pm 1.806
\]
\[
CI = (21.794, 25.406)
\]

So, the **99% confidence interval** is approximately \( (21.79, 25.41) \).

---

### Summary:
1. **90% CI for heights**: \( (70.54, 71.46) \)
2. **95% CI for heights**: \( (70.45, 71.55) \)
3. **99% CI for heights**: \( (70.28, 71.72) \)
4. **90% CI for tax form time**: \( (22.45, 24.75) \)
5. **95% CI for tax form time**: \( (22.23, 24.97) \)
6. **99% CI for tax form time**: \( (21.79, 25.41) \)

Would you like further explanation on any of these calculations?

---

### Follow-up Questions:
1. What does a confidence interval tell us about the population?
2. How does increasing the confidence level affect the width of the interval?
3. How does sample size influence the standard error and confidence interval?
4. What assumptions are necessary for constructing these confidence intervals?
5. Why do we use the z-value and how is it related to the normal distribution?

**Tip**: Increasing the confidence level widens the interval, because we need to account for more variability to be more certain.

We measured the heights of 100 Guyanese men. The mean height of men in the sample is 71 inches, with a standard deviation of 2.8 inches. Construct a 90%, 95%, and 99% confidence interval for the height of Guyanese men. A study was done to measure the time taken to complete tax forms. 100 people were surveyed, and a mean time of 23.6 hours was found to be needed to complete tax forms, with a standard deviation of 7.0 hours. Calculate the 90%, 95%, and 99% confidence intervals for the time taken to complete tax forms.

Learn how to calculate confidence intervals for the height of Guyanese men and the time taken to complete tax forms using standard statistical methods. This problem involves calculating 90%, 95%, and 99% confidence intervals based on sample data. Ideal for students in statistics courses.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation