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Hypothesis testing is a statistical method used to make inferences or draw conclusions about a population parameter based on sample data. It involves setting up two competing statements: the null hypothesis (\( H_0 \)) and the alternative hypothesis (\( H_1 \)).
- **Null Hypothesis**: This is the statement that there is no effect or no difference. In our case, it claims that the median impurity level (\( \tilde{\mu} \)) is equal to 2.5 ppm.
- **Alternative Hypothesis**: This suggests there is an effect or a difference. Here, it proposes that the median impurity level is less than 2.5 ppm.
The goal of hypothesis testing is to determine whether the sample data provides enough evidence to reject the null hypothesis. A significance level (\( \alpha \)) is set, often 0.05, which represents a 5% risk of concluding an effect exists when there is none.

- **Steps in Hypothesis Testing**: - Formulate \( H_0 \) and \( H_1 \). - Choose the significance level (\( \alpha \)). - Compute the test statistic from the sample data. - Determine the p-value. - Compare the p-value to \( \alpha \).If the p-value is less than \( \alpha \), reject \( H_0 \). Otherwise, do not reject \( H_0 \), indicating insufficient evidence to support \( H_1 \).

The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, each with the same probability of success. It is applicable when each trial has two possible outcomes: success or failure.
For a random variable \( X \) that follows a binomial distribution with \( n \) trials and probability \( p \) of success, the probability of getting exactly \( k \) successes is given by the formula:\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]In the context of our exercise, the sign test counts observations below 2.5 ppm as "success."

- **Parameters**: - \( n = 22 \) (total observations) - \( p = 0.5 \) (probability of success under \( H_0 \)) - \( k = 18 \) (observations below 2.5 ppm)
The binomial test determines the likelihood of observing \( k \) successes, helping to decide if \( H_0 \) can be rejected.

When conducting a hypothesis test with a large sample size, the binomial distribution can be approximated by a normal distribution. This is useful because the normal distribution has simpler properties and is easier to work with. The transition occurs according to the central limit theorem, which states that as the sample size \( n \) becomes large, the distribution approaches a normal shape.

In our exercise, we utilize this approximation to perform the sign test due to \( n = 22 \) not being extremely large, but sufficient for approximation.

- **Normal Approximation Formula**: - Calculate the test statistic using: \[ Z = \frac{X - n/2}{\sqrt{n/4}} \] where \( X = 18 \)The calculated Z-score is then compared against a standard normal distribution to find the corresponding p-value. If the p-value is below the selected significance level, the null hypothesis is rejected. This approach simplifies complex calculations and still delivers precise results.

Testing a hypothesis about the median involves determining if the median of a population is at a specific value. It's useful when the assumption of normality might not hold, as the median is a measure of central tendency that is less sensitive to outliers compared to the mean.

For data that are not symmetric, like impurity levels, the sign test is particularly effective. It assumes that data should symmetrically distribute around a central point under \( H_0 \). It evaluates the number of data points above and below a claimed median and employs the binomial or their normal approximation to test the claims statistically.
- **Steps for Median Testing**: - Perform a sign test by counting observations above and below the median. - Use a binomial framework with \( p = 0.5 \) to compute the p-value. - Alternatively, apply normal approximation for large samples to streamline calculations.
Ultimately, if the computed p-value is less than the significance level, we conclude that the sample median is statistically different from the proposed value.