Quality of Ceramica America Sinks

1 Quality of Ceramica America Sinks Summary of Data The proportion of sinks that "Passed” are as follows. The Cracked or Chipped test: 91.5% (0.915), the Glaze Quality test: 94% (0.94), and the Stress Test: 92% (0.92). The proportions confirm Modelo Americano sinks have high passing rates in all three quality tests. In this situation, the least percentage is 91.5%, and the highest is 94%. Thus, new Modelo Americano sinks are better quality than traditional commercial sinks, which historically had an 85% passing rate.

Graphical displays can be used to understand or explain the survey results. Bar charts visualize the proportions of sinks passing each quality test. One bar for each test can be displayed, and the height of each bar will represent the percentage of sinks passing that particular test. Pie charts are good for visualizing the distribution of sinks passing each test in percentages.

Each pie chart slice will represent the corresponding proportion.

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2 Point Estimate and Confidence Interval The point estimate of the proportion of all Modelo Americano sinks that got a "Passing" grade will be the overall passing rate calculated as the proportion of sinks with a P for all tests. = 91.5% * 94% * 92% ≈ 79.64% Calculating a 98% confidence interval for the "Passing" grade proportion follows the following formula, given that the sample size is large (200 sinks) to suggest a normal distribution for the sample proportion. Hence, Confidence Interval = Point Estimate ± (Z * Standard Error) Where: Z is the critical value for a 98% confidence level. For a two-tailed test, Z ≈ 2.33.

Standard Error = √((p * (1 - p)) / n), where p is the point estimate and n is the sample size.

Confidence Interval = 0.7964 ± (2.33 * √((0.7964 * (1 - 0.7964)) / 200)) =0.7964 ± 0.066 The 98% confidence interval for the proportion of all Modelo Americano sinks that got a "Passing" grade is approximately 73% to 86.2%. There is 98% confidence that the true proportion of all Modelo Americano sinks that passed all three tests falls within this range.

Margin of Error The margin of Error can be considered half the confidence interval's width in the following way. The margin of Error = (Upper Bound - Lower Bound) / 2. A larger sample size can decrease it to get a precise estimate and reduces the standard Error. One can also adjust the

3 confidence level. Lowering it to 95% instead of 98% will result in a narrower confidence interval with a smaller margin of Error but slightly lower confidence. The other way is to sustain the randomness and representativeness of the sample for results to be highly generalizable.

Developing a 98% Confidence Interval The proportion of sinks that passed the Cracked or Chipped test: Sample proportion (p): 0.915 (from the data) Sample size (n): 200 (from the data) Standard Error = √((p * (1 - p)) / n) = √((0.915 * (1 - 0.915)) / 200) ≈ 0 Confidence Interval = p ± (Z * Standard Error) = 0.915 ± (2.33 * 0.0197) ≈ 0.87 to 0.96 The confidence is at 98% that the true proportion of all Modelo Americano sinks that passed the Cracked or Chipped test falls within the range of approximately 87.8% to 95.2%.

Margin of Error = (Upper Bound - Lower Bound) / 2 ≈ (0.96 - 0.87) / 2 ≈ 0.045 The proportion of sinks that passed the Glaze Quality test: Sample proportion (p): 0.94 (from the data) Sample size (n): 200 (from the data).

Standard Error = √((p * (1 - p)) / n) Standard Error

4 = √((0.94 * (1 - 0.94)) / 200) ≈ 0.0167 Confidence Interval = p ± (Z * Standard Error) = 0.94 ± (2.33 * 0.0167) ≈ 0.90 to 0.979 Thus, there is 98% confidence in all Modelo Americano sinks passing the Glaze Quality test falling within the range of approximately 90% to 97.9%.

The margin of Error ≈ (0.979 - 0.90) / 2 ≈ 0.078 The proportion of sinks that passed the Stress test: Sample proportion (p): 0.92 (from the data) Sample size (n): 200 (from the data) Standard Error = √((p * (1 - p)) / n) = √((0.92 * (1 - 0.92)) / 200) ≈ 0.0142 Confidence Interval = p ± (Z * Standard Error) = 0.92 ± (2.33 * 0.0142) ≈ 0.892 to 0.948 The results are 98% confident that the true proportion of all Modelo Americano sinks passing the Stress test fall within the range of approximately 89.2% to 94.8%.

The margin of Error ≈ (0.948 - 0.892) / 2 ≈ 0.028 Hypothesis Testing Null Hypothesis (H0):

5 The proportion of all Modelo Americano sinks that pass all three tests is less than or equal to the ordinary passing rate (p ≤ 0.85).

Alternative Hypothesis (Ha): The proportion of all Modelo Americano sinks that pass all three tests exceeds the ordinary passing rate (p > 0.85).

Level of Significance (α): α = 0.02 Point Estimate (pE) = Proportion passing all three tests = 0.915 * 0.94 * 0.92 ≈ 0.7964 Standard Error = √((pE * (1 - pE)) / n) Standard Error = √((0.7964 * (1 - 0.7964)) / 200) ≈ 0.0285 Test Statistic (z) = (pE - p0) / Standard Error where p0 is the hypothesized proportion under the null hypothesis (p0 = 0.85).

Test Statistic (z) = (0.7964 - 0.85) / 0.0285 ≈ -1.8807 Using a standard normal distribution table shows that the p-value for a test statistic of 1.8807 is approximately 0.029. The null hypothesis fails since the p-value (0.0005) is more than the significance level (α = 0.02). There is a lack of evidence to confirm that the proportion of all Modelo Americano sinks that get a Pass grade is larger than the already given Pass rate.

Alternatively, with α = 0.02, the critical value for a one-tailed test is approximately 2.33 for a 98% confidence level. We accept the null hypothesis since the test statistic (-1.8807) is more than -2.33.

If there is a lower significance level (such as α = 0.01), it will be harder to reject the null hypothesis. There is less likelihood of making a Type I Error (false positive), but there is a high likelihood of a Type II error (false negative). The lower significance level increases the threshold for considering the evidence against the null hypothesis. On the other hand, if one goes with a

6 higher significance level, such as α = 0.05, it becomes easier to reject the null hypothesis. The changes increase the likelihood of making a Type I error (false positive) but lower the likelihood of a Type II error (false negative). The changes reduce the threshold for considering the evidence against the null hypothesis.

The company has additional hypothesis testing options as follows. It can do a hypothesis test for each quality test and assess if there are significant differences in passing rates for each test (Buglear, 2007). There is also an option to compare passing rates between Modelo Americano sinks and traditional commercial sinks to determine if the new sinks are better quality.

Recommendations and Magnitude of Improvement The recommendation from the substantial evidence suggests that the proportion of Modelo Americano sinks that get a "Passing" grade is larger than the traditional "Passing" rate.

Cerámica América would be to promote the Modelo Americano sinks as a higher-quality product. Doing so would help grow the customer base. The proportion passing all three tests for the Modelo Americano sinks (0.7964) is lower than the traditional passing rate (0.85). The reduction in passing rate is approximately 7.6 percentage points. However, there could be better or more reliable results if there is an increase in the sample size to get precise estimates.

Extending the study period to cover the different marketing periods is also enough. More data collection points can also help, such as customer feedback and other market sentiments.

7 References Perron, P., & Wada, T. (2016). Measuring business cycles with structural breaks and outliers: Applications to international data. Research in Economics, 70(2), 281-303.

Buglear, J. (2007). Quantitative business methods. Taylor & Francis.