Someone can help me to figure out these Math question. Appreciated!!!!!! There a

Question

Someone can help me to figure out these Math question. Appreciated!!!!!!

There are two machines available for cutting corks for wine bottles. Machine 1 produces corks with diameters that are normally distributed with mean 3 cm and standard deviation 0.1 cm. The second machine produces corks with diameters that are normally distributed with mean 3.04 cm and standard deviation 0.02 cm. Acceptable corks have diameters between 2.9 cm and 3.1 cm. Give your answers to four decimal places. A. What is the probability that a randomly chosen cork produced by Machine 1 is acceptable? B. What is the probability that a randomly chosen cork produced by Machine 2 is acceptable? 2. The National Health Statistics Reports dated Oct. 22, 2008, stated that for a sample size of 277 18-year-old American males, the sample mean waist circumference was 86.3 cm. Suppose that the population mean waist size is 85 cm and that the population standard deviation is 15 cm. How likely is it that a random sample of 277 individuals will result in a sample mean waist size of at least 86.3 cm? 3. Consider a binomial experiment with 100 trials of success probability 0.60. Use the central limit theorem to calculate the probability that the total number of successes is greater than 50.

Explanation / Answer

1.

a)

We first get the z score for the two values. As z = (x - u) / s, then as          
x1 = lower bound =    2.9      
x2 = upper bound =    3.1      
u = mean =    3      
          
s = standard deviation =    0.1      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u)/s =    -1      
z2 = upper z score = (x2 - u) / s =    1      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.158655254      
P(z < z2) =    0.841344746      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.682689492   [ANSWER]

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b)

We first get the z score for the two values. As z = (x - u) / s, then as          
x1 = lower bound =    2.9      
x2 = upper bound =    3.1      
u = mean =    3.04      
          
s = standard deviation =    0.02      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u)/s =    -7      
z2 = upper z score = (x2 - u) / s =    3      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    1.27981E-12      
P(z < z2) =    0.998650102      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.998650102   [ANSWER]  
     

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