ull and alternative hypc At least two of the popu At least two of the popu HL = "PT "C = HSA "L HpT # Hc# SA %3D %3D HL HpT Hc # SA HL = HpT= Hc = SA %3D Not all the population HL=HPT = HC = "SA %3D %3D HL= HPT = HC = SA %3D %3D Not all the population
Permutations and Combinations
If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.
Counting Theory
The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.
The second slide is the first part of the question and the first slide is the second part of the question.
![### Research Study on Job Satisfaction
A research study concluded that self-employed individuals do not experience a higher job satisfaction than individuals who are not self-employed. In this study, job satisfaction is measured using a 100-point scale. Higher scores indicate higher levels of job satisfaction.
The study involved sampling individuals from four different professions: Lawyers, Physical Therapists, Cabinetmakers, and Systems Analysts. The results are shown below for 10 individuals from each profession, with scores ranging from 18 (low satisfaction) to 86 (high satisfaction).
#### Job Satisfaction Scores
| Lawyer | Physical Therapist | Cabinetmaker | Systems Analyst |
|--------|--------------------|--------------|-----------------|
| 38 | 50 | 46 | 46 |
| 46 | 57 | 55 | 38 |
| 45 | 50 | 60 | 46 |
| 32 | 61 | 61 | 31 |
| 53 | 80 | 86 | 78 |
| 40 | 54 | 80 | 60 |
| 46 | 60 | 64 | 66 |
| 46 | 84 | 77 | 46 |
| 76 | 77 | 69 | 57 |
| 46 | 62 | 57 | 42 |
#### Hypothesis Testing
At the α = 0.05 level of significance, a test is conducted to determine if there are any differences in job satisfaction among the four professions.
- **State the null and alternative hypotheses:**
- **Option A:**
- \(H_0 : \mu_L = \mu_{PT} = \mu_C = \mu_{SA}\)
- \(H_a : \text{At least two of the population means are different}\)
- **Option B:**
- \(H_0 : \mu_L \neq \mu_{PT} = \mu_C = \mu_{SA}\)
- \(H_a : \text{Not all the population means are equal}\)
- **Option C:**
- \(H_0 : \mu_L = \mu_{PT} = \mu_C = \mu_{SA}\)
- \(H_a](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F70237df8-dfb4-41d0-af11-e6f29414ce12%2F103642e1-4816-419d-a120-db17662e7e26%2F4i15caf_processed.jpeg&w=3840&q=75)
![**Instructions for Hypothesis Testing:**
1. **Find the value of the test statistic.**
- [Input box for numerical value]
2. **Find the p-value.**
(Round your answer to three decimal places.)
- \( p \)-value = [Input box for numerical value]
3. **State your conclusion.**
- Options:
- Reject \( H_0 \). There is sufficient evidence to conclude that the mean job satisfaction rating is not the same for the four professions.
- Do not reject \( H_0 \). There is not sufficient evidence to conclude that the mean job satisfaction rating is not the same for the four professions.
- Reject \( H_0 \). There is sufficient evidence to conclude that the mean job satisfaction rating is not the same for the four professions.
- Do not reject \( H_0 \). There is not sufficient evidence to conclude that the mean job satisfaction rating is not the same for the four professions.
**Note:** Ensure you enter calculated values correctly and select the conclusion based on your analysis to accurately reflect the results of your hypothesis test.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F70237df8-dfb4-41d0-af11-e6f29414ce12%2F103642e1-4816-419d-a120-db17662e7e26%2Fwb801z6_processed.jpeg&w=3840&q=75)
![](/static/compass_v2/shared-icons/check-mark.png)
The data for 10 individuals from the four professions is given in the following table.
Lawyer | Physical Therapist | Cabinetmaker | Systems Analyst |
46 | 57 | 54 | 42 |
40 | 80 | 67 | 73 |
76 | 80 | 77 | 69 |
40 | 86 | 69 | 60 |
53 | 58 | 77 | 66 |
52 | 61 | 66 | 66 |
45 | 60 | 57 | 39 |
46 | 50 | 78 | 55 |
66 | 57 | 84 | 74 |
38 | 52 | 62 | 62 |
It is required to test whether there is any difference in job satisfaction among the four professions.
The given level of significance is 0.05.
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