A researcher wants to test the hypothesis that exposure air pollution in Toronto factory exceeds 88 hours per month. Randomly selecting 53 months, they calculated the sample mean, and they know the population standard deviation. The researchers should use the following distribution:
Student-
 distribution
normal 
-distribution

-distribution
Binomial distribution
She cannot perform the test as the population standard deviation is not available

To determine which distribution the researcher should use, let's analyze the situation:

1. **Key Information Provided:**
   - The sample size is **53 months** (large sample, \( n > 30 \)).
   - The population standard deviation is **known**.
   - The hypothesis involves testing the mean against a specified value.

2. **Key Decision Factor:**
   - If the **population standard deviation is known**, the appropriate distribution is the **normal distribution** for the test statistic.

### Answer:
The researchers should use the **normal distribution**.

### Reason:
The normal distribution is appropriate because:
- The sample size is large (\( n > 30 \)), making the Central Limit Theorem applicable.
- The population standard deviation is available, so there is no need to estimate it using the sample standard deviation (which would require the Student-t distribution).

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### Do you have any questions or want further details? Here are some follow-up questions:

1. What steps should the researcher take to set up the hypothesis test?
2. How is the test statistic calculated using the normal distribution?
3. What is the difference between the normal and Student-t distributions?
4. How does the population standard deviation influence the choice of test?
5. Can the researcher conduct the test with a smaller sample size if the standard deviation is still known?

### Tip:
Always check whether the population standard deviation is known to determine the correct distribution for hypothesis testing!

A researcher wants to test the hypothesis that exposure to air pollution in Toronto factories exceeds 88 hours per month. Randomly selecting 53 months, they calculated the sample mean, and they know the population standard deviation. The researchers should use the following distribution: Student-t distribution, normal distribution, binomial distribution, or indicate that the test cannot be performed as the population standard deviation is unavailable.

Learn how to perform hypothesis testing when the population standard deviation is known, using a normal distribution for sample means. This example tests air pollution exposure exceeding 88 hours per month over a sample of 53 months, applying key concepts such as the Central Limit Theorem and Z-test formulas. Suitable for undergraduate students.

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