Problem 10 - An Introduction to Mathematical Statistics and Its Applications

Question

A company markets two brands of latex paint regular and a more expensive brand that claims to dry an hour faster. A consumer magazine decides to test this claim by painting ten panels with each product. The average drying time of the regular brand is \(2.1\) hours with a sample standard deviation of 12 minutes. The fast-drying version has an average of \(1.6\) hours with a sample standard deviation of 16 minutes. Test the null hypothesis that the more expensive brand dries an hour quicker. Use a onesided \(H_{1}\). Let \(\alpha=0.05\).

Step-by-Step Solution

Answer

The answer will depend on the calculated test statistic and P-value. Depending on these, one would either reject or fail to reject the null hypothesis.

Step 1: Formulate the Hypotheses

The null hypothesis \(H_{0}\) is that the difference between the regular paint and the fast-drying paint is equal to one hour (since the faster brand claims to dry an hour quicker). This can be written as: \(H_{0}: \mu_{1} - \mu_{2} = 1\). The alternative hypothesis \(H_{1}\) (one-sided as stated in the task) is that the difference between the regular paint and the fast-drying paint is less than one hour, so: \(H_{1}: \mu_{1} - \mu_{2} < 1\) where \(\mu_{1}\) and \(\mu_{2}\) are the population means of the regular and fast-drying paint respectively.

Step 2: Standardize the Data

First, convert standard deviations from minutes to hours by dividing by 60. The standard deviation for the regular paint becomes \(\frac{12}{60}=0.2\) hours and for the fast-drying paint \(\frac{16}{60}\approx 0.27\) hours.

Step 3: Calculate the Test Statistic

The test statistic for a hypothesis test for the difference in two population means, with known standard deviations, is \( Z = \frac{(\bar{x_{1}} - \bar{x_{2}}) - (\mu_{1} - \mu_{2})}{\sqrt{\frac{s_{1}^{2}}{n_{1}} + \frac{s_{2}^{2}}{n_{2}}}} \) where \( \bar{x_{1}}=2.1, \bar{x_{2}}=1.6, s_{1}=0.2, n_{1}=10, s_{2}=0.27, n_{2}=10). The population means under the null hypothesis are \(\mu_{1} - \mu_{2}=1\) hour. Substituting these values, we obtain the test statistic.

Step 4: Determine the P-value

The P-value is the probability that you get a test statistic as extreme as, or more extreme than, the observed Z, assuming \(H_{0}\) is true. As we are testing a one-sided alternative hypothesis, you need to look up the probability for the observed Z in the standard normal distribution table to find the one-tail p-value.

Step 5: Interpret the Result

If the P-value is less than alpha level (\(\alpha=0.05\)), reject \(H_{0}\). This would mean that the data provides enough evidence to support the claim that the expensive paint dries quicker. If the P-value is \(\geq 0.05\), fail to reject \(H_{0}\), meaning there is not enough evidence to support the claim.

Key Concepts

Null Hypothesis

In hypothesis testing, the **null hypothesis** (denoted as \(H_{0}\)) is a statement that proposes no effect or no difference between certain characteristics of a population. It's essentially the status quo or a statement of "no change." The goal is to determine whether there is enough statistical evidence in the sample data to reject \(H_{0}\).
In our paint drying time example, the null hypothesis is formulated based on the company's claim. The claim is that the new, more expensive paint dries an hour faster than the regular paint. Therefore, our null hypothesis \(H_{0}\) is \(\mu_{1} - \mu_{2} = 1\), where \(\mu_{1}\) represents the mean drying time for the regular brand, and \(\mu_{2}\) for the fast-drying brand.
In formulating \(H_{0}\), it's vital to understand that it often acts as a claim meant to be tested. The test seeks to find contradictory evidence: that is, evidence strong enough to support the rejection of this statement. If the data does not provide enough contradiction, \(H_{0}\) is not rejected.

Alternative Hypothesis

The **alternative hypothesis** (denoted as \(H_{1}\) or \(H_{a}\)) is a statement that suggests a new effect or a difference exists. It is what you might conclude if there's enough evidence to reject the null hypothesis. In hypothesis testing, the alternative hypothesis encapsulates the claim to be tested.
For the paint drying study, the alternative hypothesis is that the difference in drying times is less than one hour. Formally, this is expressed as \(H_{1}: \mu_{1} - \mu_{2} < 1\). Here, \(\mu_{1}\) and \(\mu_{2}\) still refer to the mean drying times of the regular and fast-drying brand, respectively. This hypothesis is one-sided because it specifically states that the faster brand dries in significantly less than one hour.
Choosing the correct form of \(H_{1}\) is crucial because it defines the direction and the nature of the test. A clear understanding of the underlying scientific question will help formulate an appropriate alternative hypothesis that the data analysis can effectively target.

P-value

The **P-value** is a measure that helps determine the significance of your test results when performing hypothesis testing. It represents the probability of observing a result as extreme, or more extreme than, the one actually observed, under the assumption that the null hypothesis is true.
In simpler terms, a small P-value (typically \(<0.05\)) indicates strong evidence against the null hypothesis, and you reject \(H_{0}\). Meanwhile, a large P-value suggests that the observed data is consistent with \(H_{0}\), and you do not reject it.
For the paint exercise, since the hypothesis test uses a one-sided alternative, the P-value is determined by examining where the calculated test statistic falls within the standard normal distribution. If this P-value is less than the significance level \(\alpha = 0.05\), it suggests that the faster drying claim is statistically significant, hence the null hypothesis in this context should be rejected.
Understanding the concept of a P-value is pivotal in hypothesis testing as it quantifies the idea of statistical significance, aiding researchers in making informed decisions about their hypotheses.

Test Statistic

The **test statistic** is a standardized value that is calculated from sample data during a hypothesis test. This value is then used to decide whether to reject the null hypothesis. The specific form of the test statistic depends on the type of test being performed.
In our paint drying example, the test statistic is calculated using the sample means, sample sizes, and standard deviations of both paint types. Formally, it is given by \[ Z = \frac{(\bar{x}_{1} - \bar{x}_{2}) - (\mu_{1} - \mu_{2})}{\sqrt{\frac{s_{1}^{2}}{n_{1}} + \frac{s_{2}^{2}}{n_{2}}}} \] where \(\bar{x}_{1}\) and \(\bar{x}_{2}\) are the sample means, \(s_{1}\) and \(s_{2}\) are the sample standard deviations, and \(n_{1}\) and \(n_{2}\) are the sample sizes of the regular and fast-drying paint respectively. This calculated value shows how far the data departs from the null hypothesis.
If the test statistic value falls into the critical region as determined by the specified significance level, the null hypothesis is rejected. Thus, comprehending test statistics is essential as they transform sample data into actionable insights regarding the validity of the initial hypothesis.