Here is the scenario from week 6:While working in quality control, your job is to make sure that 95% of the 500 products which leave the factory each day are up to specifications. To test this, each day you select 25 products at random and check their quality. If no more than 1 product is poor quality, it is considered acceptable. If 2 or more products are poor quality, you must test all 500 of the products manufactured that day, a costly and time-consuming process. **Back in Quality Control!**For this portion, refer to the [Week 6 Collaborative SMP Assignment](#), the scenario with the quality control testing:- Why can we not use the z scores associated with the probabilities in these sampling distributions to answer these questions? Be specific.- Change the numbers in the scenario, to resolve the issue you just explained, and explain why these changes resolve the issue.- Sketch the sampling distributions resulting from your changed numbers and rephrase the questions in terms of sampling distributions.- Answer the questions in the scenario, with your new numbers, using the tools from this week for the sampling distribution of a sample proportion.- Illustrate the answers to the questions on the graphs of the sampling distributions.**Salamanders Again!**For this portion, refer to the [Week 6 Collaborative SMP Assignment](#), the scenario with the salamanders:- Describe the distribution of samples of size \( n = 100 \) from this population.- Sketch this sampling distribution, and also sketch the distribution of salamander masses. Compare and contrast the two distributions.- What is the probability of getting a sample of size \( n = 100 \) with mean \( \overline{X} < 7.2 \)? Explain your process.- What is the probability of getting an individual salamander with a mass of \( X < 7.2 \) grams? Explain your process.- Explain the difference between these two probabilities.

The main aim of the production process is to maintain the quality of the outgoing lot to 95%. Each day 500 products leave the factory. 25 products at randomly selected for quality checks. If no more than 1 product is poor quality, it is considered acceptable. If 2 or more products are poor quality, then all 500 of the products must be tested manufactured that day.The Sampling Distribution: For a sample of size n=25, let x denotes the acceptable number of poor quality products. it is given that x≤1. Since at least one product has to be tested, so the value of x =1. The sampling distribution of the sample proportion will be approximately normal for large sample size(>30) and when np0≥5 and n1-p0≥5, the binomial distribution can be approximated by normal distribution, and these estimates are even better when np0≥10 and n1-p0≥10 We cannot use z-scores here because the sample size is small and the assumption np0≥5 and n1-p0≥5 is not satisfied here.Changing the numbers: Let us consider a random sample of 50 products for quality check. The number of poor quality products allowed are at most 5. If no more than 5 products are poor in quality, it is considered acceptable. If 6 or more products are poor quality, you must test all 500 of the products manufactured that day, a costly and time-consuming process. Clearly, n=50, x=5 p=550=0.1 Assumption of random sample, large sample size and np0=5≥5 and n1-p0=45≥5 are satisfied. According to central limit theorem, the sampling distribution of sample proportion will be normally distributed with mean np=5 and variance np1-p=4.5 According to Empirical rule, 68%, 95%, and 99.7% of the values lie within one, two, and three standard deviations of the mean. μ±σ=5±4.5=2.879, 7.12μ±2σ=5±2×4.5=0.758, 9.242μ±3σ=5±3×4.5=-1.363, 11.36 The sketch of the graph is as follows: