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Find the test statistic and dont round answer

To calculate the test statistic for this hypothesis test, we will use the following formula for the two-sample t-test assuming unequal variances (since population standard deviations are not assumed to be equal):

\[
t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}
\]

Where:
- \(\bar{x}_1 = 0.56\) (mean reduction in pain after magnet treatment)
- \(\bar{x}_2 = 0.52\) (mean reduction in pain after sham treatment)
- \(s_1 = 0.91\) (standard deviation for magnet treatment)
- \(s_2 = 1.51\) (standard deviation for sham treatment)
- \(n_1 = n_2 = 21\) (sample sizes for both treatments)

Let's plug these values into the formula and calculate the test statistic.The test statistic, without rounding, is approximately \( t = 0.104 \).

Would you like further details or have any questions?

Here are 5 related questions for deeper understanding:
1. How is the t-distribution used when sample sizes are small?
2. What assumptions must be met to use the two-sample t-test?
3. How would the test statistic change if the sample sizes were different?
4. How does unequal variance impact hypothesis testing procedures?
5. What is the importance of the significance level in hypothesis testing?

**Tip:** When performing hypothesis tests, always check the assumptions about the data, such as normality and independence, to ensure the test results are valid.

Use a 0.05 significance level to test the claim that those treated with magnets have a greater mean reduction in pain than those given a sham treatment (similar to placebo). What is the test statistic, and don't round the answer?

Learn how to calculate the test statistic for a two-sample t-test with unequal variances using a significance level of 0.05. The problem compares pain reduction in a magnet treatment group and a sham treatment group. Suitable for undergraduate statistics students.

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