In hypothesis testing, to determine whether to reject or fail to reject the null hypothesis, we compare the p-value to the significance level \(\alpha\). In your case, the significance level is \(\alpha = 0.05\).
Here are the steps involved in the hypothesis test:
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State the Hypotheses
- Null Hypothesis (\(H_0\)): \(\mu \leq 0\) (the mean weight loss is less than or equal to 0)
- Alternative Hypothesis (\(H_a\)): \(\mu > 0\) (the mean weight loss is greater than 0)
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Calculate the Test Statistic The test statistic for a one-sample t-test can be calculated using the formula: \[ t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} \] where:
- \(\bar{x} = 2.2\) (sample mean)
- \(\mu_0 = 0\) (hypothesized mean under the null hypothesis)
- \(s = 6.1\) (sample standard deviation)
- \(n = 76\) (sample size)
Plugging the values into the formula: \[ t = \frac{2.2 - 0}{6.1 / \sqrt{76}} = \frac{2.2}{6.1 / \sqrt{76}} \] First, calculate \(6.1 / \sqrt{76}\): \[ \sqrt{76} \approx 8.7178 \] \[ 6.1 / \sqrt{76} \approx 0.699 \] Now, calculate \(t\): \[ t = \frac{2.2}{0.699} \approx 3.15 \]
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Determine the degrees of freedom The degrees of freedom for this test is: \[ df = n - 1 = 76 - 1 = 75 \]
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Find the p-value To find the p-value associated with the calculated t-statistic of approximately 3.15 with 75 degrees of freedom, you can use a t-distribution table or an online calculator.
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Comparison If the p-value is less than or equal to \(0.05\), you will reject the null hypothesis. If the p-value is greater than \(0.05\), you will fail to reject the null hypothesis.
The specific number to compare the p-value to in this case is \(0.05\).
To summarize, you compare the p-value you calculate or obtain from statistical software with \(0.05\) to determine whether to reject or fail to reject the null hypothesis.