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Hypothesis testing is a statistical method used to determine the likelihood that a given hypothesis about a dataset is true, based on sample data. The hypothesis is often tested against a null hypothesis, which represents a default or no-effect statement. In the context of our exercise, the null hypothesis (H鈧�) is that the filling machine operates randomly concerning overfilling or underfilling the bottles, while the alternative hypothesis (H鈧�) would imply non-random operation (systematic overfilling or underfilling). We start by formulating these hypotheses and proceed to test them using statistical methods.
The steps include setting a significance level ( \( \alpha \)), usually 0.05, which indicates a 5% risk of rejecting the null hypothesis when it is true. The final goal is to use data to determine whether to reject H鈧� or not.

Quality control ensures that products meet set specifications and standards. In this exercise, the quality control manager wants to verify that the bottling machine consistently fills bottles correctly. Bottles filled according to specification are marked as accepted (A), while overfilled or underfilled bottles are rejected (R). The randomness of the filling machine is tested using statistical tools to ensure that any deviations in filling volumes are due to random fluctuations rather than systematic errors.
This type of quality control is crucial in production environments to maintain product consistency, customer satisfaction, and compliance with regulatory standards.

The Z-score is a measure that describes a data point's relation to the mean of a group of values, expressed in terms of standard deviations. In our problem, we calculate the Z-score to determine how many standard deviations away the observed number of runs (R) is from the expected number ( \( \mu_R \)).
To compute the Z-score, we first calculate the mean number of runs: \[ \mu_R = \frac{2n_A n_R}{n_A + n_R} + 1 \] where \( n_A = 16 \) and \( n_R = 4 \). Thus \( \mu_R = 7.4 \).
Next, we calculate the variance: \[ \sigma_R^2 = \frac{2n_A n_R (2n_A n_R - n_A - n_R)}{(n_A + n_R)^2 (n_A + n_R - 1)} \] yielding \( \sigma_R^2 = 5.39 \). The standard deviation ( \( \sigma_R \)) is the square root of the variance, \( 2.32 \).
Finally, the Z-score formula is applied: \[ Z = \frac{R - \mu_R}{\sigma_R} \] Plugging in R = 4, we get \( Z = -1.47 \). This Z-score tells us how far the observed number of runs is from the expected, providing insight into whether the filling deviations are random.

Statistical significance helps to determine whether the observed data is due to chance or indicates a real effect. In hypothesis testing, we compare the calculated Z-score with critical values from the Z-distribution table. For a significance level ( \( \alpha \)) of 0.05 in a two-tailed test, the critical Z-values are \( \pm 1.96 \).
If the calculated Z-score falls within this range, we fail to reject the null hypothesis, indicating that there isn't sufficient evidence that the filling machine's deviations are non-random. If the Z-score exceeded \( \pm 1.96 \), it would suggest that the deviations are statistically significant, meaning the machine does not fill bottles randomly.
In our exercise, the Z-score is -1.47, which is within the critical range, so we conclude that the filling deviations are likely due to random variation, supporting the null hypothesis.