Question

In: Statistics and Probability

Information for Problems 1 - 5: Photoresist is a light sensitive material used in industrial processes...

Information for Problems 1 - 5: Photoresist is a light sensitive material used in industrial processes like photolithography and photoengraving to form a patterned coating on a surface. There are two types of photoresists: positive and negative. A positive photoresist is one in which the portion of the photoresist that is exposed to light becomes soluble to the photoresist developer. A negative photoresist is one in which the portion of the photoresist that is exposed to light becomes insoluble to the photoresist developer. Suppose the thickness, x, of photoresist applied to wafers in semiconductor manufacturing at a particular location on the wafer is uniformly distributed between 0.2150 and 0.2350 micrometers.

1.     Find the probability distribution function for x.  

2.     Find the cumulative distribution function of photoresist thickness.

3.     Determine the proportion of wafers that exceed 0.2275 micrometers in photoresist thickness.

4.     What is the minimum thickness that is exceeded by 30% of the wafers?   

5.     Determine the mean of the photoresist thickness.

Information for Problems 6 - 10: Cholesterol is a fatty substance that is an important part of the outer lining (membrane) of cells in the body of animals. Its normal range for an adult is 120-240 mg/dl. The Food and Nutrition Institute (FNI) of the Philippines found that the total cholesterol level for Filipino adults has a mean, µ, of 148.9 mg/dl. Furthermore, the FNI reports that 92% of adults have a total cholesterol level below 200 mg/dl and that total cholesterol level is normally distributed.

6.     What is the standard deviation of this normal distribution? Give answer to 1 decimal places.

7.     An adult is considered to be at high risk for cardiovascular disease if his/her total cholesterol level is above 240.8 mg/dl. What proportion of Filipino adults are at high risk for cardiovascular disease? Give answer to 4 decimal places.

8.     An adult is considered to be at moderate risk for cardiovascular disease if his/her total cholesterol level is between 200 and 240.8 mg/dl. What proportion of Filipino adults are at moderate risk for cardiovascular disease?   Give answer to 4 decimal places.

9.     What is the value of total cholesterol level that exceeds 90% of the Filipino population? Give answer to 2 decimal places.

10. Above what total cholesterol value will 25% of the Filipino population fall? Give answer to 2 decimal places.

Information for Problems 11 - 15: A fiber spinning process currently produces a fiber whose strength is normally distributed with a mean of 71 N/m2. The minimum acceptable strength is 65N/m2. Give answer to 4 decimal places.

11. Ten percent of the fiber produced by the current method fails to meet the minimum specification. What is the standard deviation of the fiber strengths in the current process?   

12. If the mean remains at 71N/m2 and the minimum acceptable strength remains at 65N/m2, what must the standard deviation be so that only 5% of the fiber will fail to meet the specification?

13. If the mean remains at 71N/m2 and the minimum acceptable strength remains at 65N/m2, what must the standard deviation be so that only 1% of the fiber will fail to meet the specification?

14. If the standard deviation is 3N/m2, to what value must the mean be set so that only 1% of the fiber will fail to meet the minimum specification of 65N/m2?

15. If the standard deviation is 3N/m2, to what value must the mean be set so that only 2% of the fiber will fail to meet the specification of 65N/m2?       

Solutions

Expert Solution

Given :

1) The probability distribution function is given by:

hence,

2. The cumulative distribution function of X is given by:

3. The proportion of wafers that exceeds 0.2275 micrometers in photoresist thickness is given by:

4. The minimum thickness that is exceeds by 30% of the wafers is given by:

The minimum thickness that is exceeds by 30% of the wafers is 0.2290.

5. The mean of the photoresist thickness is given by:

6:

According to range rule the standard deviation is approximately equal to one fourth of the range of the data so

7:

The z-score for X = 240.8 is

The proportion of adults at hig risk is

8:

The z-score for X = 200.8 is

The required proprotion is:

9:

Here we need z-score that has 0.10 area to its left. The z-score -1.28 has 0.10 area to its left. So required cholesterol level is:

10:

Here we need z-score that has 0.25 area to its right and 0.75 area to its left. The z-score 0.675 has 0.75 area to its left. The requried cholesterol level is


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