Problem
The research department of a household appliances manufacturing company has devised a bimetallic thermal sensor for a toaster. According to the research department, this new sensor will reduce the return of appliances with a year-full warranty by 2%-6%. To validate this claim, two groups of toasters were selected by the testing department: a group produced with new sensors and a group produced with old sensors. Both of these were exposed to a worth of wear of an average year. Of the 250 toasters manufactured with new sensors, 8 would be returned. Of the 250 toasters manufactured with old sensors, 17 would be returned. A statistical procedure is needed for the research department’s claim to be to verified or refuted.
Theory
Since one needs to verify or refute a hypothesis, it is only reasonable to resort to hypothesis testing for the solution to this problem. According to Casella and Berger (2021), a hypothesis is a population parameter statement. The purpose of hypothesis testing is to determine which of the two complementary hypotheses is true based on a population sample. These complementary hypotheses are referred to as the null hypothesis and the alternative hypothesis and are respectively designated as H₀ and H₁.
Z-Test Model
For solving this particular problem, one is to use the Z test model for differences in proportions. As per Schumacker (2017), the Z-test questions may include the difference between a one-sample percentage and a determinate population percentage or the difference in the proportion of populations between two independent groups. Other types of questions may be related to differences in the proportions of populations between related groups. Z-test statistics can provide answers to research questions related to a single percentage, percentage differences between independent groups, and dependent percentage differences.
Z-Test Assumptions
When it comes to Two Sample Z Proportion Hypothesis Tests, there are some assumptions about those. According to Hessing (n.d.), data is both populations’ simple random values, both populations follow the Gaussian Probability distribution, and samples are never dependent of one another. Moreover, as per Hessing (n.d.), two sample Z proportion tests have their hypotheses. The null hypothesis and the alternative hypothesis have been mentioned before. In addition to that, there are right-tailed and left-tailed hypotheses: this is when the difference between proportions of the population is greater or smaller than the expected difference, respectively.
Solution
The solution starts with the definition of the null and alternative hypotheses. Null hypothesis means the proportions are the same, and alternative means they are not. Old sensors returns = x₁ = 8, and new sensors returns = x₂ = 17. Old sensors = n₁ = 250, and new sensors = n₂ = 250. Old sensors return = p̂₁=17/250= 0.068 = 6,8%, and new sensors return = p̂₂=8/250 = 0.032 = 3.2%. First compute p₀ = x₁+x₂ /n₁+n₂ = 8+17/250+250=25/500=0.05 = 5%.
Then, for the final Z formula, a couple more figures are needed. First, there is p̂1 – p̂2 = 0.068 – 0.032 = 0.036. Then, there is p0 * (1 – p0 ) = 0.05 * (1 – 0.05) = 0.05 * 0.95 = 0.0475. Finally, there is (1/n₁) + (1/n₂) = (1/250) + (1/250) = 0.004 + 0.004 = 0.008. Therefore, Z = (0.036)/ SQRT ( (0.0475) * (0.008)). Z = (0.036)/ SQRT ( 0.00038)) = (0.036)/ (0.019493) = 1.84
Flowchart
A flowchart, a diagram depicting a process, workflow, or algorithm, can be used to show the process of solving a two-proportions Z-test. This particular flowchart shows the operations that were necessary to calculate the final percentage difference between the two groups. All one needed to know was the number of old sensors and new sensors, and the number of old sensors and new sensor returns, as well as the relevant formulas.
Summary
In conclusion, the percentage difference between the groups of old sensor and new sensor returns equals 1.84. It shows a significant difference between the two product lines: the proportions are not the same. Therefore, the null hypothesis is rejected. When it comes to the research department’s claim, it is refuted since 1.84 is less than 2, and 2 percent was the lower limit of expected results.
References
Casella, G., & Berger, R. L. (2021). Statistical inference. Cengage Learning.
Hessing, T. (n.d.). Two sample Z tests for proportions. Six Sigma Study Guide.
Schumacker, R. E. (2017). Learning statistics using R. SAGE Publications.