5. A random sample of 4500 U.S. employees was chosen and asked whether the "play hooky", that is, call in sick at least once a year when they simply need to relax; 1080 responded "yes". Use this information to set up a 95% confidence interval for the proportion, p, of all U.S. employees who play hooky. 1080 0.24 4500 4: 1-.24= .76 95% C.I 2c= O124 I 4500 1.96 %3D (0.275,0.2515)
Unitary Method
The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products.
Speed, Time, and Distance
Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.
Profit and Loss
The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.
Units and Measurements
Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.
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![**Estimating the Proportion of U.S. Employees Who "Play Hooky"**
A random sample of 4500 U.S. employees was chosen, and each was asked whether they "play hooky" — that is, call in sick at least once a year when they simply need to relax. Out of these, 1080 employees responded “yes”. Using this data, we will set up a 95% confidence interval for the proportion, \( p \), of all U.S. employees who played hooky.
### Step-by-Step Calculation:
1. **Sample Proportion (\(\hat{p}\))**: Calculate the sample proportion of employees who answered "yes".
\[
\hat{p} = \frac{x}{n} = \frac{1080}{4500} = 0.24
\]
2. **Complement of the Sample Proportion (\(\hat{q}\))**: This is obtained by subtracting the sample proportion from 1.
\[
\hat{q} = 1 - \hat{p} = 1 - 0.24 = 0.76
\]
3. **Critical Value (\(Z_c\))**: For a 95% confidence interval, the critical value is determined from the standard normal distribution (Z-table).
\[
Z_c = 1.96
\]
4. **Standard Error (SE)**:
\[
SE = \sqrt{\frac{\hat{p} \cdot \hat{q}}{n}} = \sqrt{\frac{0.24 \cdot 0.76}{4500}}
\]
5. **Margin of Error (ME)**: Multiply the critical value by the standard error.
\[
ME = Z_c \cdot SE = 1.96 \cdot \sqrt{\frac{0.24 \cdot 0.76}{4500}}
\]
6. **Confidence Interval**: Add and subtract the margin of error from the sample proportion to obtain the confidence interval.
\[
\hat{p} \pm ME \implies 0.24 \pm 0.0125
\]
7. The calculated interval is:
\[
(0.2275, 0.2525)
\]
### Interpretation:
With 95% confidence, the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1bc805eb-60ed-457b-ab8e-02c516698c74%2Fb26505bd-caf3-4a98-bb11-e6c4ead05d6b%2Fz52wcr_processed.jpeg&w=3840&q=75)
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