In a simple random sample of size 62, there were 37 individuals in the category of interest. (a) Compute the sample proportion *p*. Round the answer to at least three decimal places. The sample proportion is 0.597. are the assumptions for a hypothesis test satisfied? Explain. , the number of individuals in each category is 10.

To determine whether the assumptions for a hypothesis test are satisfied, we need to check certain criteria, especially when dealing with proportions. These assumptions are as follows:

### Assumptions for Hypothesis Test for Proportion
1. **Random Sample**: The sample must be randomly selected from the population. This assumption is satisfied because the problem states that it is a simple random sample.
   
2. **Independence**: The observations in the sample must be independent of each other. This is typically satisfied if the sample size is less than 10% of the population. We aren't given the population size, but we assume it's large enough for this condition to hold.

3. **Normal Approximation**: For the sampling distribution of the sample proportion to be approximately normal, two conditions should be met:
   - \( np \geq 10 \)
   - \( n(1 - p) \geq 10 \)

   Here:
   - \( n = 62 \) (sample size)
   - \( p = 0.597 \) (sample proportion)

Let's compute these values:

- \( np = 62 \times 0.597 = 37.014 \)
- \( n(1 - p) = 62 \times (1 - 0.597) = 62 \times 0.403 = 24.986 \)

Both \( np \) and \( n(1 - p) \) are greater than 10. Therefore, the normal approximation condition is satisfied.

### Conclusion:
Since all the necessary assumptions are met (random sample, independence, and normal approximation), the conditions for conducting a hypothesis test for the sample proportion are satisfied.

Do you need more details on how hypothesis testing works or further clarifications?

Here are five follow-up questions you could explore:
1. What is the null hypothesis for this hypothesis test?
2. How do you compute the standard error for the sample proportion?
3. What is the alternative hypothesis for this test?
4. How would you calculate the test statistic?
5. How do you interpret the p-value in hypothesis testing?

**Tip:** Always verify if both \( np \) and \( n(1 - p) \) are greater than 10 before using the normal approximation in hypothesis testing for proportions.

In a simple random sample of size 62, there were 37 individuals in the category of interest. (a) Compute the sample proportion *p*. Round the answer to at least three decimal places. The sample proportion is 0.597. Are the assumptions for a hypothesis test satisfied? Explain. The number of individuals in each category is 10.

Explore the computation of sample proportion from a random sample of size 62, with 37 individuals in the category of interest. Learn whether the assumptions for hypothesis testing are satisfied, including random sampling, independence, and normal approximation. Suitable for students in Grades 11-12 or introductory college-level statistics.

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