Problem 39 - An Introduction to Mathematical Statistics and Its Applications

Question

Below are the lengths (in \(\mathrm{mm}\) ) of furniture dowels recorded as part of an ongoing quality-control program. Listed are the measurements made on thirty samples (each of size 4) taken in order from the assembly line. Is the variation in the sample averages random with respect to the number of runs up and down? Do an appropriate hypothesis test at the \(\alpha=0.05\) level of significance. | Sample | y_(1) | y_(2) | y_(3) | y_(4) | bar(y) | | :---: | :---: | :---: | :---: | :---: | :---: | | 1 | 46.1 | 44.4 | 45.3 | 44.2 | 45.0 | | 2 | 46.0 | 45.4 | 42.5 | 44.4 | 44.6 | | 3 | 44.3 | 44.0 | 45.4 | 43.9 | 44.4 | | 4 | 44.9 | 43.7 | 45.2 | 44.8 | 44.7 | | 5 | 43.0 | 45.3 | 45.9 | 43.8 | 44.5 | | 6 | 46.0 | 43.2 | 44.4 | 43.7 | 44.3 | | 7 | 46.0 | 44.6 | 45.4 | 46.4 | 45.6 | | 8 | 46.1 | 45.5 | 45.0 | 45.5 | 45.5 | | 9 | 42.8 | 45.1 | 44.9 | 44.3 | 44.3 | | 10 | 45.0 | 46.7 | 43.0 | 44.8 | 44.9 | | 11 | 45.5 | 44.5 | 45.1 | 47.1 | 45.6 | | 12 | 45.8 | 44.6 | 44.8 | 45.1 | 45.1 | | 13 | 45.1 | 45.4 | 46.0 | 45.4 | 45.5 | | 14 | 44.6 | 43.8 | 44.2 | 43.9 | 44.1 | | 15 | 44.8 | 45.5 | 45.2 | 46.2 | 45.4 | | 16 | 45.8 | 44.1 | 43.3 | 45.8 | 44.8 | | 17 | 44.1 | 44.8 | 46.1 | 45.5 | 45.1 | | 18 | 44.5 | 43.6 | 45.1 | 46.9 | 45.0 | | 19 | 45.2 | 43.1 | 46.3 | 46.4 | 45.3 | | 20 | 45.9 | 46.8 | 46.8 | 45.8 | 46.3 | | 21 | 44.0 | 44.7 | 46.2 | 45.4 | 45.1 | | 22 | 43.4 | 44.6 | 45.4 | 44.4 | 44.5 | | 23 | 43.1 | 44.6 | 44.5 | 45.8 | 44.5 | | 24 | 46.6 | 43.3 | 45.1 | 44.2 | 44.8 | | 25 | 46.2 | 44.9 | 45.3 | 46.0 | 45.6 | | 26 | 42.5 | 43.4 | 44.3 | 42.7 | 43.2 | | 27 | 43.4 | 43.3 | 43.4 | 43.5 | 43.4 | | 28 | 42.3 | 42.4 | 46.6 | 42.3 | 43.4 | | 29 | 41.9 | 42.9 | 42.0 | 42.9 | 42.4 | | 30 | 43.2 | 43.5 | 42.2 | 44.7 | 43.4 |

Step-by-Step Solution

Answer

The procedure includes hypothesis formulation, conduction of the run test and result interpretation. Given a hypothetical test calculation, the results would be based on the comparison of the calculated Z-value with the critical Z value. We then either reject or do not reject the null hypothesis based on where our test statistic lies in reference to critical values ±1.96.

Step 1: Formulate Hypotheses

Null hypothesis \(H_0\): The variation in the sample averages is random (there is no pattern related to the number of runs).\nAlternative hypothesis \(H_a\): The variation in the sample averages is not random (there is a pattern related to the number of runs).

Step 2: Conduct hypothesis test

We will use a run test for randomness, which is suitable for this type of problem, where we are testing if the sample values are occurring randomly or not.\nWe first list the sample averages in order. Then, we make a column next to the ordered averages and mark if the next sample average is higher (+) or lower (-) than the previous one. The number of runs is the number of times we change from + to - or vice-versa.\nAfter counting the number of runs, we can calculate the test statistic Z, which is the number of observed runs minus the expected number of runs, divided by the standard deviation of the number of runs. A run is defined as a sequence of increasing or decreasing values.\nThe observed number of runs, the expected number of runs, and the standard deviation can be calculated using the following formulas:\nExpected runs \(E(R) = \frac{2n_1*n_2}{n} + 1\)\nVariance of runs \(\sigma^2(R) = \frac{2n_1*n_2(2n_1*n_2-n)}{n^2(n-1)}\)\nWhere \(n_1\) is the number of positive runs, \(n_2\) is the number of negative runs and \(n\) is the total number of runs.\nThe test statistic is given by \(Z = \frac{R - E(R)}{\sqrt{\sigma^2(R)}}\) where R is the observed number of runs.\nThe Z-value will then be compared with the critical Z value for a two-tailed test at the 0.05 significance level. The critical Z value for 0.05 significance level is approximately ±1.96.

Step 3: Draw Conclusion

We reject the null hypothesis if the test statistic Z is less than -1.96 or greater than +1.96. If the test statistic falls between -1.96 and +1.96, we do not reject the null hypothesis.\nRejecting the null hypothesis would suggest that there is a significant pattern related to the number of runs and the sample averages are not random. Not rejecting the null hypothesis would suggest that the variation in the sample averages is indeed random. The interpretation is based on the alpha level of 0.05, suggesting a 5% chance we are making an error if we reject the null hypothesis.

Key Concepts

Hypothesis Testing

Hypothesis testing is a crucial statistical method used to make decisions about a population based on a sample. In this process, we start by making an initial assumption called the null hypothesis, denoted as \(H_0\). For this exercise, the null hypothesis is that the variation in the sample averages is random, implying no detectable pattern in the sequence of runs. On the other side, we have the alternative hypothesis, denoted by \(H_a\), suggesting that the variation is not random and there's a significant pattern. The primary aim is to determine whether there is enough evidence to reject the null hypothesis in favor of the alternative. We achieve this by calculating a test statistic and comparing it to a critical value determined by the significance level, which in this case is \( \alpha = 0.05 \). If our test statistic falls within the critical region, we reject the null hypothesis. This method helps statisticians and quality control experts to gauge the reliability of patterns found in data and supports decision-making in various fields like manufacturing. A clear understanding of hypothesis testing empowers accurate conclusions that are statistically valid.

Run Test for Randomness

A run test for randomness is a non-parametric test used to analyze a sequence of data points, checking for randomness between upward or downward moves. It's essential in scenarios where the order of data, rather than the data points themselves, is of interest. The test involves partitioning the data set into runs, which are sequences of identical elements or observations. For example, if the data set has alternating segments of highs and lows, each segment is a run. To conduct a run test on this exercise, we first ordered the sample averages and marked whether each subsequent value increased (+) or decreased (-) compared to the previous one. Each continuous series of + or - forms a run. The vital step in a run test is calculating the number of observed runs in this sequence. The run test provides a framework to calculate expected runs \(E(R)\) and variance \(\sigma^2(R)\), helping us find the test statistic \(Z\). Comparing the calculated Z statistic against a critical value tells us if the sequence deviates significantly from randomness. This method is widely applied in quality control and financial analysis to detect anomalies or patterns that might not be apparent at first glance.

Quality Control Sampling

Quality control sampling is an essential practice in manufacturing and production fields, ensuring products meet specified standards. By taking small, manageable samples from a production line, businesses can draw insights about the overall process and identify whether it operates within acceptable parameters. In this exercise, samples of furniture dowels were collected in batches, with their lengths measured to assess the variation in the process. Using statistical tests like the run test for randomness can help identify whether observed variations are random or indicative of a systematic issue. Implementing sampling and randomness tests in quality control helps in:

• Ensuring consistent product quality.
• Detecting production flaws early on.
• Making informed decisions on process improvements.

Regular quality control sampling supports the maintenance of high standards, resulting in customer satisfaction and reduced costs due to minimized defects. By relying on statistical assessments, businesses can safeguard their reputation and maintain competitive edges.