The thickness of a plastic film (in mils) on a substrate...

The thickness of a plastic film (in mils) on a substrate material is thought to be influenced by the temperature at which the coating is applied. In completely randomized experiment, 11 substrates are coated at $125^{\circ} \mathrm{F}$, resulting in a sample mean coating thickness of $\bar{x}_{1}=103.5$ and a sample standard deviation of $s_{1}=10.2$. Another 13 substrates are coated at $150^{\circ} \mathrm{F}$ for which $\bar{x}_{2}=99.7$ and $s_{2}=20.1$ are observed. It was originally suspected that raising the process temperature would reduce mean coating thickness. (a) Do the data support this claim? Use $\alpha=0.01$ and assume that the two population standard deviations are not equal. Calculate an approximate $P$ -value for this test. (b) How could you have answered the question posed regarding the effect of temperature on coating thickness by using a confidence interval? Explain your answer.

The thickness of a plastic film (in mils) on a substrate material is thought to be influenced by the temperature at which the coating is applied. In completely randomized experiment, 11 substrates are coated at $125^{\circ} \mathrm{F}$, resulting in a sample mean coating thickness of $\bar{x}_{1}=103.5$ and a sample standard deviation of $s_{1}=10.2$. Another 13 substrates are coated at $150^{\circ} \mathrm{F}$ for which $\bar{x}_{2}=99.7$ and $s_{2}=20.1$ are observed. It was originally suspected that raising the process temperature would reduce mean coating thickness.
(a) Do the data support this claim? Use $\alpha=0.01$ and assume that the two population standard deviations are not equal. Calculate an approximate $P$ -value for this test.
(b) How could you have answered the question posed regarding the effect of temperature on coating thickness by using a confidence interval? Explain your answer.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample of data to infer that a certain condition holds for the entire population. It involves formulating a null hypothesis (which typically represents no effect or no difference) and an alternative hypothesis (which reflects the effect or difference considered of interest), then using a test statistic to infer whether the observed data are extreme under the assumption that the null hypothesis is true.

Two-Sample t-Test (Welch's t-test)

The two-sample t-test compares the means of two independent groups to determine if there is a statistically significant difference between them. When the assumption of equal population variances does not hold, the Welch's t-test is used instead, which adjusts the degrees of freedom in the test statistic to account for the unequal variances of the two groups.

P-value

A P-value is the probability of obtaining test results at least as extreme as the ones observed during the study, assuming that the null hypothesis is true. It is used to decide whether to reject the null hypothesis; a smaller P-value indicates stronger evidence in favor of the alternative hypothesis, with a predefined significance level (alpha) serving as the threshold for this decision.

Confidence Intervals

A confidence interval provides a range of values that is likely to contain the true difference between population parameters (such as means) with a certain degree of confidence (typically 95% or 99%). It offers an alternative way of statistical inference by allowing researchers to assess the effect size and the precision of the estimate, where intervals not containing the value of no difference (such as zero) indicate a statistically significant result.

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