Suppose that a researcher is interested in whether an exercise improves intelligence. The researcher randomly selects 100 participants, assigns them to an exercise program, and measures intelligence at the end of the exercise program. The measurement of intelligence has a known or assumed average score of μ = 100 and a known or assumed standard deviation of σ = 15 in the population.
The researcher should use a      hypothesis test to evaluate the treatment affected intelligence.
If σ and σ2
2
were unknown, the researcher should use a     .

To evaluate the effect of the treatment on intelligence, the researcher should use a **z-test** when the population standard deviation (σ) is known.

If σ (the population standard deviation) and σ² (the population variance) were unknown, the researcher should use a **t-test** instead.

Would you like a more detailed explanation of why the z-test and t-test are appropriate in these cases?

Here are 5 related questions to expand your understanding:

1. What is the primary difference between a z-test and a t-test?
2. How does sample size affect the choice between a z-test and a t-test?
3. What is the critical value in hypothesis testing, and how is it determined for a z-test?
4. How would the researcher interpret the p-value in this scenario?
5. What assumptions must be met to perform a t-test?

**Tip:** When the sample size is large (usually over 30), the t-distribution approaches the z-distribution. Therefore, for large samples, even with an unknown standard deviation, results of the t-test are similar to the z-test.

Suppose that a researcher is interested in whether an exercise improves intelligence. The researcher randomly selects 100 participants, assigns them to an exercise program, and measures intelligence at the end of the exercise program. The measurement of intelligence has a known or assumed average score of μ = 100 and a known or assumed standard deviation of σ = 15 in the population. The researcher should use a hypothesis test to evaluate the treatment affected intelligence. If σ and σ² were unknown, the researcher should use a.

Learn how to apply hypothesis testing to determine whether an exercise improves intelligence. This problem involves selecting between a z-test and t-test based on known and unknown population parameters (σ and σ²). Understand key concepts such as the Central Limit Theorem, and explore how the sample size influences test choice.

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