A sample of 25 different payroll departments found that the employees worked an average of 310.3 days a year with a standard deviation of 23.8 days. What is the 90% confidence interval for the average days worked, u, of all payroll departments. A) 301.0 < < 319.6 B) 298.0 < < 322.6 C) 302.4 < < 318.2 D) 314.1 < < 316.8
Compound Probability
Compound probability can be defined as the probability of the two events which are independent. It can be defined as the multiplication of the probability of two events that are not dependent.
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Probability theory is a branch of mathematics that deals with the subject of probability. Although there are many different concepts of probability, probability theory expresses the definition mathematically through a series of axioms. Usually, these axioms express probability in terms of a probability space, which assigns a measure with values ranging from 0 to 1 to a set of outcomes known as the sample space. An event is a subset of these outcomes that is described.
Conditional Probability
By definition, the term probability is expressed as a part of mathematics where the chance of an event that may either occur or not is evaluated and expressed in numerical terms. The range of the value within which probability can be expressed is between 0 and 1. The higher the chance of an event occurring, the closer is its value to be 1. If the probability of an event is 1, it means that the event will happen under all considered circumstances. Similarly, if the probability is exactly 0, then no matter the situation, the event will never occur.
![**Problem Statement:**
A sample of 25 different payroll departments found that the employees worked an average of 310.3 days a year with a standard deviation of 23.8 days. What is the 90% confidence interval for the average days worked, \( \mu \), of all payroll departments?
**Options:**
A) \( 301.0 < \mu < 319.6 \)
B) \( 298.0 < \mu < 322.6 \)
C) \( 302.4 < \mu < 318.2 \)
D) \( 314.1 < \mu < 316.8 \)
**Explanation:**
To find the 90% confidence interval for the average days worked, you can use the formula for the confidence interval of a sample mean:
\[ \bar{x} \pm Z \left(\frac{\sigma}{\sqrt{n}}\right) \]
Where:
- \( \bar{x} \) is the sample mean, which is 310.3 days.
- \( \sigma \) is the standard deviation, which is 23.8 days.
- \( n \) is the sample size, which is 25.
- \( Z \) is the Z-value for a 90% confidence level, which is approximately 1.645.
Using these values, you determine the correct confidence interval from the given options.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F98f4f7b9-70b3-4d15-8afc-3e235681bb40%2F2faaef4c-ca7f-46c7-998c-10a29e0c870f%2Fdg4wqpd_processed.jpeg&w=3840&q=75)
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