Problem 3 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

Table 27.5 lists the results of tensile adhesion tests on 22 U-700 alloy specimens. The data are loads at failure in MPa. The sample mean is \(13.71\) and the sample standard deviation is \(3.55\). You may assume that the data originated from a normal distribution with expectation \(\mu\). One is interested in whether the load at failure exceeds \(10 \mathrm{MPa}\). We investigate this by means of a \(t\)-test for the null hypothesis \(H_{0}: \mu=10\). a. What do you choose as the alternative hypothesis? b. Compute the value of the test statistic and report your conclusion, when Derformina the test at level \(0.05\)

Step-by-Step Solution

Answer

Reject the null hypothesis; the load at failure exceeds 10 MPa.

Step 1: Define the Hypotheses

We are going to perform a one-sample t-test. Our null hypothesis is \( H_0: \mu = 10 \), and since we want to know if the mean load at failure exceeds 10 MPa, the alternative hypothesis is \( H_a: \mu > 10 \).

Step 2: Compute the Test Statistic

The test statistic for a one-sample t-test is given by \( t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}} \), where \( \bar{x} = 13.71 \) is the sample mean, \( \mu_0 = 10 \) is the hypothesized mean under the null hypothesis, \( s = 3.55 \) is the sample standard deviation, and \( n = 22 \) is the sample size. Plugging in the values, we get: \[ t = \frac{13.71 - 10}{3.55/\sqrt{22}} \approx \frac{3.71}{0.756} \approx 4.91 \]

Step 3: Determine the Critical Value and Conclusion

With a one-tailed t-test at a significance level of \( \alpha = 0.05 \) and \( n - 1 = 21 \) degrees of freedom, we look up the critical value in a t-distribution table. The critical value for \( t \) at \( \alpha = 0.05 \) for 21 degrees of freedom is approximately 1.721. Since the calculated t-value (4.91) is greater than the critical value (1.721), we reject the null hypothesis.

Key Concepts

One-Sample t-Test

The one-sample t-test is a statistical method used to determine whether the mean of a single sample is equal to a known value or a theoretical expectation. In this context, we are checking if the average load at failure of U-700 alloy specimens exceeds a specific benchmark, 10 MPa. The test involves comparing the sample mean to the hypothesized population mean using sample data.

• Sample Mean (\( \bar{x} \)) – this is the average value obtained from the sample, in our case, 13.71 MPa.
• Hypothesized Mean (\( \mu_0 \)) – this is the mean value we are testing against, here it is 10 MPa.

By calculating the t-statistic, we can decide if there's enough evidence to say that there is a significant difference between the sample mean and the hypothesized mean. If this difference is too large to attribute to random variation, the null hypothesis can be rejected.

Hypothesis Testing

Hypothesis testing is a core method in statistics used to make inferences about a population. It's conducted by setting up two opposing hypotheses:

• Null Hypothesis (\( H_0 \)): This claims no effect or no difference, stating that the population mean is equal to a specific value, which in our case is 10 MPa.
• Alternative Hypothesis (\( H_a \)): This represents what you're trying to prove. It suggests that the population mean exceeds the hypothesized value, which is depicted as \( \mu > 10 \) MPa.

The process involves using sample data to calculate a test statistic, which then determines the likelihood of observing the data assuming the null hypothesis is true. Based on this test statistic, we either reject the null hypothesis or fail to provide evidence against it, taking into account a pre-set level of significance or alpha level, usually set at 0.05.

Critical Value

The critical value is a threshold that helps to decide whether to reject the null hypothesis in a hypothesis test. It depends on the significance level (\( \alpha \)) and the degrees of freedom from the sample size. For this exercise:

• Confidence Level: We are performing the test at a significance level of 0.05, which means we are 95% confident in our results and allow a 5% risk of wrongly rejecting the null hypothesis.
• Degrees of Freedom: This is calculated as the sample size minus one (\( n - 1 \)), so with 22 samples, we have 21 degrees of freedom.
• Lookup: Using a t-distribution table, we find the critical value for \( \alpha = 0.05 \) and 21 degrees of freedom is approximately 1.721.

If our calculated t-statistic exceeds this critical value, we conclude that the evidence is strong enough to reject the null hypothesis and suggest that the true mean load exceeds 10 MPa.

t-Distribution

The t-distribution is a statistical distribution that is used when the sample size is small, and the population standard deviation is unknown. It is similar to the normal distribution but has thicker tails. These thicker tails account for the additional uncertainty introduced by estimating the population standard deviation from the sample.

• Shape: The shape of the t-distribution depends on the degrees of freedom (df). As df increases, the t-distribution approaches a normal distribution.
• Use: In our one-sample t-test, the degrees of freedom are based on the sample size, which was 21 for our test.

This distribution is crucial for determining critical values and helps to establish a decision rule regarding our hypotheses. The t-distribution is a flexible statistical tool for drawing meaningful inferences from sample data when conditions of normality are not perfectly met.

Sample Statistics

Sample statistics are numerical values that provide estimations about the population based on sample data. In the case of our t-test, they are used to calculate the t-statistic and include:

• Sample Mean (\( \bar{x} \)): The average value from our sample, providing an estimate of the central tendency, 13.71 MPa in our case.
• Sample Standard Deviation (\( s \)): This measures the dispersion in the sample data. For our test, it was calculated to be 3.55 MPa, suggesting the extent of variation around the sample mean.
• Sample Size (\( n \)): The number of observations or data points in our sample, which in this exercise is 22.

These statistics form the foundation for the t-test calculation, allowing us to construct the t-statistic: \( t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}} \).This formula forms the basis for making inferences about the population based on our sample.