Hypothesis Testing

Week 14 (Tutorial)

For the tutorial session in week 14, we were introduced to hypothesis testing.

Background Information

Hypothesis testing refers to the formal procedures used by experimenters or researchers to accept or reject statistical hypotheses.

• Step 1: State the statistical hypotheses. This involves stating the null hypothesis (H0) and the alternative hypothesis (H1). The null hypothesis states that differences in sample observations result purely from chance and that there is no statistically-significant difference in a set of given observations. The alternative hypothesis states that differences in sample observations are statistically-significant and influenced by some non-random causes.
• Step 2: Formulate an analysis plan. There are two commonly used statistical tools, which is the t-test and the z-test. The t-test is usually used when the sample size is lesser than 30, and the z-test is used when the sample size is greater than or equal to 30. At this juncture, we will also decide if a test is one-tailed (when sign in H1 is ≠) or two-tailed (when sign in H1 is < or >).
• Step 3: Calculate the test statistic. We will select the appropriate formula to calculate the test statistic based on sample data.

Figure 1: Formulae to Use

• Step 4: Make a decision based on results. We will interpret the results of the tests and make a decision on accepting or rejecting the null hypothesis.

The ideal way to determine if a statistical hypothesis is true is by examining the entire population. Since that is often impractical, researchers usually examine only a random sample from the population. If the sample data is not consistent with the statistical hypothesis, the hypothesis is rejected.

There are trade-offs when one uses sample data to estimate the properties of a population since the sample may not be representative of the population, resulting in decision errors. Two types of errors can result from hypothesis testing. Type I Error occurs when the researcher rejects a null hypothesis when it is true. Type II Error occurs when the researcher fails to reject a null hypothesis that is false.

For this blog entry, I was required to use the data collected for FULL Factorial and FRACTIONAL Factorial from the Design of Experiments (DOE) Practical session to perform hypothesis testing on. 

My DOE Practical team members are:

1. Steward (Iron Man): Run #2 from FRACTIONAL factorial and Run #2 from FULL factorial
2. Wayne (Thor): Run #3 from FRACTIONAL factorial and Run #3 from FULL factorial
3. Jiayu (Captain America): Run #5 from FRACTIONAL factorial and Run #5 from FULL factorial
4. Xinni (Black Widow): Run #8 from FRACTIONAL factorial and Run #8 from FULL factorial
5. Nick (Hulk): Run #3 from FRACTIONAL factorial and Run #3 from FULL factorial
6. - (Hawkeye): Run #8 from FRACTIONAL factorial and Run #8 from FULL factorial

Figure 2: Data collected for FULL factorial design using CATAPULT A

Figure 3: Data collected for FRACTIONAL factorial design using CATAPULT B 

The Question (Purpose of conducting the hypothesis testing): The catapult (the ones that were used in the DOE practical) manufacturer needs to determine the consistency of the products they have manufactured. Therefore, they want to determine whether CATAPULT A produces the same flying distance of projectile as that of CATAPULT B.

The scope of the test: The human factor is assumed to be negligible. Therefore, different users will not have any effect on the flying distance of projectile.

Flying distance for catapult A and catapult B is collected using the factors below:

• Arm length = 32 cm
• Start angle = 0 degrees
• Stop angle = 55 degrees

Step 1: State the Statistical Hypotheses

Null Hypothesis (H0): Catapult A and Catapult B will produce the same flying distance. (μ1 = μ2)

Alternative Hypothesis (H1): Catapult A and Catapult B will produce different flying distances. (μ1 ≠ μ2)

Step 2: Formulate an Analysis Plan

Sample size is 8 (<30) Therefore, a t-test will be used.

Since the sign of H1 is ≠, a two tailed test is used.

Significance level (α) used in this test is 0.05. The value of 0.05 is the default significance level to be used.

Step 3: Calculate the Test Statistic

State the mean and standard deviation of sample catapult A

Figure 4: Data for Catapult A

State the mean and standard deviation of sample catapult B:

Figure 5: Data for Catapult B

Compute the value of the test statistic (t):

Figure 6: Calculation of t

Step 4: Make a Decision based on Results

Type of test: Two-tailed test, thus critical value tα/2 = 0.05/2 = ± 0.025

1 – 0.025 = 0.975

From the t-distribution table, t0.975 = ±2.145

 

Figure 7: Comparing Values of t and critical values

Since the value of -1.50252645 is within the acceptance range, H0 is accepted.

Step 5: Conclusion That Answers the Initial Question

At 0.05 level of significance, Catapult A and Catapult B will produce the same flying distance.

Step 6: Compare Your Conclusion with the Conclusion from the other Team Members

The conclusion gathered from the other team members is that there is no difference in the flying distance from both catapults. Thus, it can be inferred that the consistency of the products manufactured is high, and there are no quality control issues between different catapults of the same model.

Reflection

In general, I feel that hypothesis skill is a very important skill to master, as it can often be the deciding factor in whether a hypothesis can be deemed as correct or not. Initially, when we had just first learned about this topic, I was quite confused and overwhelmed, as there were many new concepts that we had just been introduced to. However, after attempting some practice questions, I got the hang of it, and am now able to better appreciate hypothesis testing as a whole. I believe that this skill will be able to help me in my future projects, especially in the upcoming Capstone Project.