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According to the article "Modeling and Predicting the Effects of Submerged Arc Weldment Process Parameters on Weldment Characteristics and Shape Profiles" ( \(J\). of Engr. Manuf., 2012: 1230-1240), the submerged arc welding (SAW) process is commonly used for joining thick plates and pipes. The heat affected zone (HAZ), a band created within the base metal during welding, was of particular interest to the investigators. Here are observations on depth (mm) of the HAZ both when the current setting was high and when it was lower. \(\begin{array}{llllll}\text { Non-high } & 1.04 & 1.15 & 1.23 & 1.69 & 1.92 \\\ & 1.98 & 2.36 & 2.49 & 2.72 & \\ & 1.37 & 1.43 & 1.57 & 1.71 & 1.94 \\ & 2.06 & 2.55 & 2.64 & 2.82 & \\ \text { High } & 1.55 & 2.02 & 2.02 & 2.05 & 2.35 \\ & 2.57 & 2.93 & 2.94 & 2.97 & \end{array}\) a. Construct a comparative boxplot and comment on interesting features. b. Is it reasonable to use the two-sample \(t\) test to test hypotheses about the difference between true average HAZ depths for the two conditions? c. Does it appear that true average HAZ depth is larger for the higher current condition than for the lower condition? Carry out a test of appropriate hypotheses using a significance level of .01.

Step by step solution

01

Organize the Data

First, we organize the given data into two groups: Non-high current setting and High current setting. For the Non-high setting, list the depths: 1.04, 1.15, 1.23, 1.69, 1.92, 1.98, 2.36, 2.49, 2.72, 1.37, 1.43, 1.57, 1.71, 1.94, 2.06, 2.55, 2.64, 2.82. For the High setting, list the depths: 1.55, 2.02, 2.02, 2.05, 2.35, 2.57, 2.93, 2.94, 2.97.

02

Create Boxplots

With the data organized, create boxplots for each group. The boxplot will show the median, quartiles, and any potential outliers. Place both boxplots on the same scale to allow for comparison.

03

Comment on Boxplot Features

Examine features such as the median, range, and any apparent skewness. Also, note any outliers or differences in spread between the two sets.

04

Check Conditions for Two-Sample t-test

Verify assumptions for performing a two-sample t-test: independence of samples, normality of the data, and equal variances. Use normal probability plots or histograms to check normality. Employ Levene's Test or compare variances to check equal variance assumption.

05

Set Hypotheses for the t-test

State the null hypothesis (\(H_0: \mu_1 = \mu_2\)) which asserts that the true average depths are equal for the two conditions, and the alternative hypothesis (\(H_a: \mu_1 < \mu_2\)) that posits higher average depth for the high current setting.

06

Perform Two-Sample t-test

Calculate the t-statistic and p-value by using the sample means, sample variances, and sample sizes. Since we are testing at a 0.01 significance level, we will compare the p-value to this threshold to decide whether to reject the null hypothesis.

07

Interpret Test Results

If the p-value is less than 0.01, reject the null hypothesis indicating the true average HAZ depth is larger for the high current condition. Otherwise, we do not have sufficient evidence to reject the null hypothesis.

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Key Concepts

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

Boxplot Analysis

When we're dealing with data, it's often useful to visualize it using different methods to make patterns and comparisons clear. A boxplot, sometimes called a box-and-whisker plot, is one of these methods. It's a simple but effective way to display the distribution of a dataset and to easily compare different groups.

• The box in the boxplot arises from the middle 50% of the data, which represents the interquartile range (IQR).
• A line inside the box marks the median, the middle point of the data.
• Whiskers extend from the box to the smallest and largest values, considering values within 1.5 * IQR from the quartiles.

In our exercise, the boxplots for the high and non-high current settings allow us to quickly see how the depths of the heat-affected zone compare. Look at the medians and the spread of the data. If one box is notably higher or the whiskers are much longer, it can signal differences in the variabilities or typical values between the groups.

Null and Alternative Hypotheses

In hypothesis testing, we begin with the null hypothesis, which is an assumption that there is no difference or effect. It's basically a statement of "no change." In contrast, the alternative hypothesis proposes that there is a difference or effect.

• The null hypothesis denoted as (\( H_0: \, \mu_1 = \mu_2 \)) suggests that the true average HAZ depths are the same under both current settings.
• The alternative hypothesis (\( H_a: \, \mu_1 < \mu_2 \)) states that the true average depth for the high current setting is greater than that of the non-high setting.

These hypotheses are tested statistically, and the outcome dictates whether it's plausible to reject the null hypothesis in favor of the alternative one. This process is vital to ensuring scientific investigations are robust and conclusions drawn are justified.

Test of Equal Variances

Before conducting a two-sample t-test, it's crucial to understand if the assumption of equal variances holds. When the variances of two populations are equal, the statistical power and accuracy of the test improve. One way to test this assumption is through a test like Levene's Test.

• Levene's Test assesses the null hypothesis that the variances in two or more groups are equal.
• We're looking to see if there is no significant difference in variances \((H_0: \, \sigma_1^2 = \sigma_2^2)\).
• If p-value from Levene's Test is below a certain significance level, it casts doubt on the equal variances assumption.

If variances are unequal, a modification of the t-test called Welch's t-test could be more appropriate, as it does not assume equal variances between the groups.

Significance Level in Hypothesis Testing

In hypothesis testing, the significance level, also known as the alpha level, is the threshold used to decide whether to reject the null hypothesis. It's essentially the probability of making a Type I error, which is the incorrect rejection of the true null hypothesis.

• A typical alpha level is 0.05, but in our exercise, a stricter 0.01 level is used, indicating a 1% risk of incorrectly rejecting the null hypothesis.
• This strict threshold is beneficial in critical fields where the cost of errors is high.
• If the p-value obtained from the test is below the alpha level, the results are considered statistically significant, warranting rejection of the null hypothesis.

Using a significance level provides clarity and consistency in decision-making, ensuring that only compelling evidence leads to rejecting the null hypothesis.