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Chapter 14: Problem 11

A modification has been made to the process for producing a certain type of "time-zero" film (film that begins to develop as soon as a picture is taken). Because the modification involves extra cost, it will be incorporated only if sample data strongly indicates that the modification has decreased true average developing time by more than \(1 \mathrm{~s}\). Assuming that the developing-time distributions differ only with respect to location if at all, use the Wilcoxon rank-sum test at level \(.05\) on the accompanying data to test the appropriate hypotheses. \(\begin{array}{lllllllll}\text { Original } & & & & & & & & & \\ \text { Process } & 8.6 & 5.1 & 4.5 & 5.4 & 6.3 & 6.6 & 5.7 & 8.5 \\\ \begin{array}{l}\text { Modified } \\ \text { Process }\end{array} & 5.5 & 4.0 & 3.8 & 6.0 & 5.8 & 4.9 & 7.0 & 5.7\end{array}\)

Short Answer

Expert verified
Use the Wilcoxon rank-sum test and compare the statistic to the critical value to determine if the modification reduces time by more than 1 second.

Step by step solution

01

State the Hypotheses

We will use the Wilcoxon rank-sum test to compare the two processes. The null hypothesis \( H_0 \) is that there is no decrease or the decrease in true average developing time is \( \leq 1 \text{ s} \). The alternative hypothesis \( H_a \) is that the modified process decreases the true average developing time by more than 1 s.
02

Calculate the Ranks

Combine and sort all observation times from both groups. Then, assign ranks to the data, averaging ranks where values are the same. Calculate the ranks for each observation in both the original and modified processes.
03

Sum Ranks for Each Sample

Based on the ranks calculated, sum up the ranks separately for the original process and the modified process. Calculate \( R_1 \) for the original process observations and \( R_2 \) for the modified process observations.
04

Determine Critical Value

Consult the Wilcoxon rank-sum distribution table to determine the critical value for \( n_1 = 8, n_2 = 8 \) at the \( \alpha = 0.05 \) significance level. For a one-sided test, the critical value will determine the threshold for rejecting the null hypothesis.
05

Compare Test Statistic to Critical Value

Determine the test statistic, typically the smaller of the two rank sums \( R_1 \) and \( R_2 \), or transform using known formulas to compare against critical values if needed. Compare the result to the critical value found in Step 4 to decide whether to reject the null hypothesis.
06

Make a Conclusion

If the test statistic is less than the critical value, reject the null hypothesis. If it is not, fail to reject the null hypothesis. This will indicate whether the modification decreases the developing time by more than 1 second or not.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Hypothesis Testing
Hypothesis testing is a statistical method employed to determine whether there is enough evidence in a sample to infer that a certain condition is true for an entire population.
This process involves formulating two statements:
  • **Null Hypothesis (H鈧�):** It represents the status quo or a baseline measurement. In the context of our example, the null hypothesis is that the modification either does not decrease the mean developing time or does so by no more than 1 second.
  • **Alternative Hypothesis (H鈧�):** It indicates the opposite of the null hypothesis, suggesting what we aim to prove. Here, it states that the modification leads to a decrease in the average developing time by more than 1 second.
These hypotheses are tested using sample data.
The goal is to see if there is enough evidence to reject the null hypothesis in favor of the alternative hypothesis. It鈥檚 crucial to carefully define these hypotheses before embarking on any statistical test, as they guide the entire testing process.
Exploring Non-Parametric Statistics
Non-parametric statistics are methods that do not rely on data belonging to any particular distribution.
This is particularly useful when dealing with real-world data that does not precisely fit normal distribution assumptions.
The Wilcoxon rank-sum test is one such non-parametric test.
It is used when comparing two independent samples, similar to the t-test, but without needing the assumption of normality.
  • **Robustness:** Non-parametric tests are less sensitive to outliers and can be more robust in the presence of non-homogeneous data.
  • **Data Requirements:** They often require fewer assumptions, making them suitable for ordinal data or data on an interval or ratio scale that violates normality.
The strength of non-parametric methods like the Wilcoxon rank-sum test lies in their ability to handle data variability with fewer preconditions. This makes them versatile tools for hypothesis testing in diverse data scenarios.
Determining the Significance Level
The significance level, typically denoted as \(\alpha\), is a threshold set by the researcher to determine the probability of incorrectly rejecting a true null hypothesis.
A common choice for \(\alpha\) is 0.05, which implies a 5% risk of erroneously declaring a result significant.
  • **Setting \(\alpha\) Value:** It represents the chance you are willing to take of making a Type I error (rejecting the true null hypothesis). A lower \(\alpha\) indicates a stricter criterion for accepting significant results.
  • **Comparative Nature:** The test statistic calculated from your data is compared against a critical value determined by \(\alpha\) to decide the significance of the results. If the test statistic falls under the critical threshold, the null hypothesis is rejected.
Choosing an appropriate significance level is fundamental to research and provides a benchmark for assessing the reliability of your conclusions.
It dictates how strict the test is in deeming results statistically significant, impacting the study's power to detect actual effects.

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Most popular questions from this chapter

The accompanying data resulted from an experiment to compare the effects of vitamin \(\mathrm{C}\) in orange juice and in synthetic ascorbic acid on the length of odontoblasts in guinea pigs over a 6-week period ("The Growth of the Odontoblasts of the Incisor Tooth as a Criterion of the Vitamin C Intake of the Guinea Pig," J. Nutrit., 1947: 491-504). Use the Wilcoxon rank-sum test at level \(.01\) to decide whether true average length differs for the two types of vitamin \(C\) intake. Compute also an approximate \(P\)-value. [Hint: See Exercise 14.] \(\begin{array}{lrrrrrr}\text { Orange Juice } & 8.2 & 9.4 & 9.6 & 9.7 & 10.0 & 14.5 \\ & 15.2 & 16.1 & 17.6 & 21.5 & & \\ \text { Ascorbic Acid } & 4.2 & 5.2 & 5.8 & 6.4 & 7.0 & 7.3 \\ & 10.1 & 11.2 & 11.3 & 11.5 & & \end{array}\)

The accompanying 25 observations on fracture toughness of base plate of \(18 \%\) nickel maraging steel were reported in the article "Fracture Testing of Weldments" (ASTM Special Publ. No. 381, 1965: 328-356). Suppose a company will agree to purchase this steel for a particular application only if it can be strongly demonstrated from experimental evidence that true average toughness exceeds 75. Assuming that the fracture toughness distribution is symmetric, state and test the appropriate hypotheses at level \(.05\), and compute a \(P\)-value. $$ \begin{array}{lllllllll} 69.5 & 71.9 & 72.6 & 73.1 & 73.3 & 73.5 & 74.1 & 74.2 & 75.3 \\ 75.5 & 75.7 & 75.8 & 76.1 & 76.2 & 76.2 & 76.9 & 77.0 & 77.9 \\ 78.1 & 79.6 & 79.7 & 80.1 & 82.2 & 83.7 & 93.7 & & \end{array} $$

Let \(X\) have the beta distribution on \([0,1]\) with parameters \(\alpha=v_{1} / 2\) and \(\beta=v_{2} / 2\), where \(v_{1} / 2\) and \(v_{2} / 2\) are positive integers. Define \(Y=\) \((X / \alpha) /[(1-X) / \beta]\). Show that \(Y\) has the \(F\) distribution with degrees of freedom \(v_{1}, v_{2}\).

The model for the data from a randomized block experiment for comparing \(I\) treatments was \(X_{i j}=\mu+\alpha_{i}+\beta_{j}+\varepsilon_{i j}\), where the \(\alpha\) 's are treatment effects, the \(\beta\) 's are block effects, and the \(\varepsilon\) 's were assumed normal with mean 0 and variance \(\sigma^{2}\). We now replace normality by the assumption that the \(\varepsilon\) 's have the same continuous distribution. A distribution-free test of the null hypothesis of no treatment effects, called Friedman's test, involves first ranking the observations in each block separately from 1 to \(I\). The rank average \(\bar{R}_{i}\) is then calculated for each of the \(I\) treatments. If \(H_{0}\) is true, the expected value of each rank average is \((I+1) / 2\). The test statistic is $$ F_{r}=\frac{12 J}{I(I+1)} \sum\left(\bar{R}_{i}-\frac{I+1}{2}\right)^{2} $$ For even moderate values of \(J\), the test statistic has approximately a chi- squared distribution with \(I-1 \mathrm{df}\) when \(H_{0}\) is true. The article "Physiological Effects During Hypnotically Requested Emotions" (Psychosomatic Med., 1963: 334-343) reports the following data \(\left(x_{i j}\right)\) on skin potential in millivolts when the emotions of fear, happiness, depression, and calmness were requested from each of eight subjects. $$ \begin{array}{lllrl} \text { Fear } & 23.1 & 57.6 & 10.5 & 23.6 \\ \text { Happiness } & 22.7 & 53.2 & 9.7 & 19.6 \\ \text { Depression } & 22.5 & 53.7 & 10.8 & 21.1 \\ \text { Calmness } & 22.6 & 53.1 & 8.3 & 21.6 \\ & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} \\ \cline { 2 - 5 } \text { Fear } & 11.9 & 54.6 & 21.0 & 20.3 \\ \text { Happiness } & 13.8 & 47.1 & 13.6 & 23.6 \\ \text { Depression } & 13.7 & 39.2 & 13.7 & 16.3 \\ \text { Calmness } & 13.3 & 37.0 & 14.8 & 14.8 \\ \hline \end{array} $$ Use Friedman's test to decide whether emotion has an effect on skin potential.

Laplace's rule of succession says that if there have been \(n\) Bernoulli trials and they have all been successes, then the probability of a success on the next trial is \((n+1) /(n+2)\). For the derivation Laplace used a beta prior with \(a=1\) and \(b=1\) for binomial data, as in Example 14.7. a. Show that, if \(a=1\) and \(b=1\) and there are \(n\) successes in \(n\) trials, then the posterior mean of \(p\) is \((n+1) /(n+2)\). b. Explain (a) in terms of total successes and failures; that is, explain the result in terms of two prior trials plus \(n\) later trials. c. Laplace applied his rule of succession to compute the probability that the sun will rise tomorrow using 5000 years, or \(n=1,826,214\) days of history in which the sun rose every day. Is Laplace's method equivalent to including two prior days when the sun rose once and failed to rise once? Criticize the answer in terms of total successes and failures.
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