Z Test For The Mean

91 Z Test For The Meanz Test For The Meanproblemyou Are The Manager

Identify the core assignment question or prompt and clean it by removing any non-essential information, repetitive lines, and extraneous instructions. The core task involves performing a Z-test for the mean and a t-test for the difference in means based on provided data, hypotheses, and significance levels, then interpreting the results to determine whether to reject or fail to reject the null hypothesis.

From the provided fragmented data and context, the main objectives are to:

  • Conduct a Z-test for the mean, using sample size, sample mean, population standard deviation, and significance level.
  • Compare the sample mean to the hypothesized population mean (368 grams).
  • Calculate the test statistic, critical values, and p-value.
  • Make a decision to reject or not reject the null hypothesis based on these calculations.
  • Similarly, for the comparison of two means, perform a pooled variance t-test for the difference in means assuming equal variances, interpret the test statistic, critical value, p-value, and arrive at a conclusion.

Paper For Above instruction

In quality assurance and process monitoring within manufacturing organizations, statistical hypothesis testing plays a critical role in ensuring product specifications are consistently met. In this scenario, Oxford Cereals employs statistical analysis to verify whether the average weight of cereal boxes aligns with the company’s specifications, which requires an average weight of 368 grams per box, at a significance level of 0.05. By implementing statistical tests such as the Z-test for the mean and t-tests for comparing two means, managers can make informed decisions about process control and quality assurance.

Performing the Z-test for the mean

Given a sample size of 25 cereal boxes with a sample mean weight of 372.5 grams, a known population standard deviation of 15 grams, and a hypothesized mean weight of 368 grams, the initial step involves calculating the standard error of the mean (SEM). The SEM is derived using the formula:

SEM = σ / √n = 15 / √25 = 15 / 5 = 3 grams

This measure indicates the expected variability of the sample mean if the process is in control. The Z-test statistic is then computed as:

Z = (x̄ - μ₀) / SEM = (372.5 - 368) / 3 ≈ 4.5 / 3 = 1.5

Using the standard normal distribution, the critical values at a 0.05 significance level for a two-tailed test are:

Lower critical value: NORMSINV(0.025) ≈ -1.96

Upper critical value: NORMSINV(0.975) ≈ 1.96

The computed Z-value of 1.5 lies within the non-rejection region between -1.96 and 1.96, indicating that there is insufficient evidence to reject the null hypothesis that the mean cereal weight is 368 grams. The p-value associated with Z=1.5 is approximately 0.1336, which exceeds the significance level of 0.05, reaffirming the conclusion that the sample does not significantly differ from the specified population mean.

Comparing the Means of Two Different Suppliers

In assessing whether two different suppliers produce cereal boxes with significantly different mean weights, a pooled variance t-test is employed, especially when the sample variances are similar, and sample sizes are reasonable. Suppose sample data from two suppliers are as follows:

  • Supplier A: n₁=17, x̄₁=19.5 grams, s₁²=16.5
  • Supplier B: n₂=24, x̄₂=20.8 grams, s₂²=19.5

The hypotheses for this test are:

  • Null hypothesis (H₀): μ₁ - μ₂ = 0 (no difference in means)
  • Alternative hypothesis (H₁): μ₁ - μ₂ ≠ 0 (there is a difference)

First, calculate the pooled variance:

pooled variance, s_p² = [ (n₁ - 1)  s₁² + (n₂ - 1)  s₂² ] / (n₁ + n₂ - 2)


= [ (17 - 1)16.5 + (24 - 1)19.5 ] / (17 + 24 - 2)


= [ 1616.5 + 2319.5 ] / 39


= [ 264 + 448.5 ] / 39 ≈ 712.5 / 39 ≈ 18.27

Next, compute the standard error (SE) of the difference in means:

SE = sqrt( s_p² * (1/n₁ + 1/n₂) ) 


= sqrt(18.27 * (1/17 + 1/24)) 


≈ sqrt(18.27 * (0.0588 + 0.0417)) 


≈ sqrt(18.27 * 0.1005) 


≈ sqrt(1.837) ≈ 1.355

The t-test statistic is then:

t = (x̄₁ - x̄₂) / SE = (19.5 - 20.8) / 1.355 ≈ -1.3 / 1.355 ≈ -0.959

Degrees of freedom: df = n₁ + n₂ - 2 = 39. Using the t-distribution table, the critical t-value for a two-tailed test at α=0.05 is approximately ±2.022. Since |-0.959|

Furthermore, the p-value associated with t = -0.959 and df=39 is approximately 0.34, indicating no significant difference between the two suppliers’ mean weights. This statistical evidence suggests that both suppliers are producing cereal boxes within comparable weight ranges, with no meaningful variation.

Implications and Conclusions

The above statistical analyses demonstrate the importance of hypothesis testing in quality management systems. The Z-test confirms that the sample mean weight does not significantly deviate from the target mean of 368 grams, supporting the continuation of current processes. Similarly, the t-test shows that there is no significant difference in means between two suppliers' products, indicating consistency among supply sources. These insights assist managers in making informed decisions regarding process controls, supplier evaluations, and quality standards.

Furthermore, these tests underscore the relevance of choosing appropriate statistical methods—Z-tests when population standard deviations are known and sample sizes are large, and t-tests when variances are unknown or sample sizes are small. Proper application of these techniques ensures that quality assurance processes are statistically valid, thus maintaining product standards and customer satisfaction.

References

  • Bluman, A. G. (2017). Elementary Statistics: A Step-by-Step Approach. McGraw-Hill Education.
  • Glen, S. (2018). Understanding Hypothesis Testing. Statistik Guru. Retrieved from https://statisticsglossary.com/hypothesis-testing
  • Montgomery, D. C., & Runger, G. C. (2014). Applied Statistics and Probability for Engineers. Wiley.
  • Walpole, R. E., Myers, R. H., Myers, S. L., & Ye, K. (2012). Probability and Statistics for Engineers and Scientists. Pearson.
  • DiCiccio, T. J., & Efron, B. (1996). Bootstrap confidence intervals. Statistical Science, 11(3), 189-228.
  • Ott, R. L., & Longnecker, M. (2010). An Introduction to Statistical Methods and Data Analysis. Brooks/Cole.
  • Upton, M. (2017). Statistical Process Control and Quality Improvement. International Journal of Production Research, 55(16), 4823-4832.
  • Zar, J. H. (2010). Biostatistical Analysis. Pearson.
  • Dean, A., & Voss, D. (2014). Design and Analysis of Experiments. Springer.
  • Jain, R. K., & Tan, V. (2017). Application of Statistical Methods in Manufacturing. Journal of Quality Technology, 49(1), 1-15.

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