Question 1)

Consider the following context:

The nutrition and food science division of Farnsworth Pharmaceuticals is alpha testing an OTC diagnostic kit so that adults who self-report a certain food allergy can affirm their insensitivity to that food allergen. Preliminary clinical trials yielded that 25% of the volunteers who suspected they had food allergies were clinically identified as sensitive or intolerant to the food allergen; unfortunately and disappointingly, only 10% of those who tested positive in the clinic also tested positive with the kit. Furthermore, of those self-reporting adult volunteers who had a negative clinic outcome, 5% tested positive anyway with the kit. [COMMENTS & HINTS: Consider clinical sensitivity to the food allergen a positive result for testing purposes. If necessary, round off percentages to the nearest hundredth, expressing as #.##% or ##.##%.]

- How many complete branches, from start to finish, would a tree diagram that models this context include????? HINT: Each branch addresses both phases of the experiment.

Question 2)

Consider the following context:

The nutrition and food science division of Farnsworth Pharmaceuticals is alpha testing an OTC diagnostic kit so that adults who self-report a certain food allergy can affirm their insensitivity to that food allergen. Preliminary clinical trials yielded that 25% of the volunteers who suspected they had food allergies were clinically identified as sensitive or intolerant to the food allergen; unfortunately and disappointingly, only 10% of those who tested positive in the clinic also tested positive with the kit. Furthermore, of those self-reporting adult volunteers who had a negative clinic outcome, 5% tested positive anyway with the kit. [COMMENTS & HINTS: Consider clinical sensitivity to the food allergen a positive result for testing purposes. If necessary, round off percentages to the nearest hundredth, expressing as #.##% or ##.##%.]

What is, therefore, the chance of a true positive outcome of this compound event: BOTH a positive result on the clinical diagnosis AND a positive result on the OTC kit diagnosis, given that the clinical diagnosis was positive?????? HINT: What is the appropriate arithmetic operation to compute the probability of a compound event that utilizes the both ... and ... conjunction? Express the answer as a percentage without insignificant digits, with percent symbol.

Question 3)

Consider the following context:

The nutrition and food science division of Farnsworth Pharmaceuticals is alpha testing an OTC diagnostic kit so that adults who self-report a certain food allergy can affirm their insensitivity to that food allergen. Preliminary clinical trials yielded that 25% of the volunteers who suspected they had food allergies were clinically identified as sensitive or intolerant to the food allergen; unfortunately and disappointingly, only 10% of those who tested positive in the clinic also tested positive with the kit. Furthermore, of those self-reporting adult volunteers who had a negative clinic outcome, 5% tested positive anyway with the kit. [COMMENTS & HINTS: Consider clinical sensitivity to the food allergen a positive result for testing purposes. If necessary, round off percentages to the nearest hundredth, expressing as #.##% or ##.##%.]

If 10% of the results from the OTC diagnostic kits are positive after the adult volunteers already tested positive with the clinical diagnosis, then what was the complementary probability for those OTC diagnostic kits that end up with negative results???? Express the answer as a percentage without insignificant digits, with percent symbol.

Question 4)

Consider the following context:

The nutrition and food science division of Farnsworth Pharmaceuticals is alpha testing an OTC diagnostic kit so that adults who self-report a certain food allergy can affirm their insensitivity to that food allergen. Preliminary clinical trials yielded that 25% of the volunteers who suspected they had food allergies were clinically identified as sensitive or intolerant to the food allergen; unfortunately and disappointingly, only 10% of those who tested positive in the clinic also tested positive with the kit. Furthermore, of those self-reporting adult volunteers who had a negative clinic outcome, 5% tested positive anyway with the kit. [COMMENTS & HINTS: Consider clinical sensitivity to the food allergen a positive result for testing purposes. If necessary, round off percentages to the nearest hundredth, expressing as #.##% or ##.##%.]

What is, therefore, the chance of a false negative outcome of this compound event: BOTH a positive result on the clinical diagnosis AND a negative result on the OTC diagnostic kit, given that the result of the clinical diagnosis was positive for food allergies?????? HINTS: As a reminder, if 10% of those who tested positive with the clinical diagnosis also tested positive with the kit, what percentage surprisingly tested negative with the kit? Utilize that percentage here in your compound probability calculations. What is the appropriate arithmetic operation to compute the probability of a compound event that utilizes the both ... and ... conjunction? Express the answer as a percentage without insignificant digits, with percent symbol.

Question 5)

Consider the following context:

The nutrition and food science division of Farnsworth Pharmaceuticals is alpha testing an OTC diagnostic kit so that adults who self-report a certain food allergy can affirm their insensitivity to that food allergen. Preliminary clinical trials yielded that 25% of the volunteers who suspected they had food allergies were clinically identified as sensitive or intolerant to the food allergen; unfortunately and disappointingly, only 10% of those who tested positive in the clinic also tested positive with the kit. Furthermore, of those self-reporting adult volunteers who had a negative clinic outcome, 5% tested positive anyway with the kit. [COMMENTS & HINTS: Consider clinical sensitivity to the food allergen a positive result for testing purposes. If necessary, round off percentages to the nearest hundredth, expressing as #.##% or ##.##%.]

If 25% of the positive results from the clinical diagnoses ended up as positive result of food sensitivity or intolerance, then what was the complementary probability for those adult volunteers who ended up with a negative result from the clinical diagnosis, apparently having no food insensitivies or allergies????? Express the answer as a percentage without insignificant digits, with percent symbol.

Question 6)

Consider the following context:

The nutrition and food science division of Farnsworth Pharmaceuticals is alpha testing an OTC diagnostic kit so that adults who self-report a certain food allergy can affirm their insensitivity to that food allergen. Preliminary clinical trials yielded that 25% of the volunteers who suspected they had food allergies were clinically identified as sensitive or intolerant to the food allergen; unfortunately and disappointingly, only 10% of those who tested positive in the clinic also tested positive with the kit. Furthermore, of those self-reporting adult volunteers who had a negative clinic outcome, 5% tested positive anyway with the kit. [COMMENTS & HINTS: Consider clinical sensitivity to the food allergen a positive result for testing purposes. If necessary, round off percentages to the nearest hundredth, expressing as #.##% or ##.##%.]

What is, therefore, the chance of a false positive outcome of the compound event: BOTH a negative result on the clinical diagnosis AND a positive result on the OTC diagnostic kit, given that the clinincal diagnosis was negative????? HINT: What is the appropriate arithmetic operation to compute the probability of a compound event that utilizes the both ... and ... conjunction? Express the answer as a percentage without insignificant digits, with percent symbol.

Question 7)

Consider the following context:

The nutrition and food science division of Farnsworth Pharmaceuticals is alpha testing an OTC diagnostic kit so that adults who self-report a certain food allergy can affirm their insensitivity to that food allergen. Preliminary clinical trials yielded that 25% of the volunteers who suspected they had food allergies were clinically identified as sensitive or intolerant to the food allergen; unfortunately and disappointingly, only 10% of those who tested positive in the clinic also tested positive with the kit. Furthermore, of those self-reporting adult volunteers who had a negative clinic outcome, 5% tested positive anyway with the kit. [COMMENTS & HINTS: Consider clinical sensitivity to the food allergen a positive result for testing purposes. If necessary, round off percentages to the nearest hundredth, expressing as #.##% or ##.##%.]

If 5% of the OTC diagnostic kit results are positive anyway following negative clinical diagnoses for food allergies, then what was the complementary probability for those that end up with a negative result from the OTC diagnostic kit after the initial negative result from the clinical testing???? Express the answer as a percentage without insignificant digits, with percent symbol.

Question 8)

Consider the following context:

The nutrition and food science division of Farnsworth Pharmaceuticals is alpha testing an OTC diagnostic kit so that adults who self-report a certain food allergy can affirm their insensitivity to that food allergen. Preliminary clinical trials yielded that 25% of the volunteers who suspected they had food allergies were clinically identified as sensitive or intolerant to the food allergen; unfortunately and disappointingly, only 10% of those who tested positive in the clinic also tested positive with the kit. Furthermore, of those self-reporting adult volunteers who had a negative clinic outcome, 5% tested positive anyway with the kit. [COMMENTS & HINTS: Consider clinical sensitivity to the food allergen a positive result for testing purposes. If necessary, round off percentages to the nearest hundredth, expressing as #.##% or ##.##%.]

What is, therefore, the chance of a true negative outcome of this compound event: BOTH a negative result on the clincial diagnosis AND a negative result on the OTC diagnostic kit, given that the clinical diagnosis result was negative???? HINT: What is the appropriate arithmetic operation to compute the probability of a compound event that utilizes the both ... and ... conjunction? Express the answer as a percentage without insignificant digits, with percent symbol.

Question 9)

Consider the following context:

The nutrition and food science division of Farnsworth Pharmaceuticals is alpha testing an OTC diagnostic kit so that adults who self-report a certain food allergy can affirm their insensitivity to that food allergen. Preliminary clinical trials yielded that 25% of the volunteers who suspected they had food allergies were clinically identified as sensitive or intolerant to the food allergen; unfortunately and disappointingly, only 10% of those who tested positive in the clinic also tested positive with the kit. Furthermore, of those self-reporting adult volunteers who had a negative clinic outcome, 5% tested positive anyway with the kit. [COMMENTS & HINTS: Consider clinical sensitivity to the food allergen a positive result for testing purposes. If necessary, round off percentages to the nearest hundredth, expressing as #.##% or ##.##%.]

Given that the self-reporting adult did indeed test positive on the OTC diagnostic kit for food sensitivity, what is the chance the adult volunteer had a positive result on the clinical diagnosis in the first place?????? HINT: Create a ratio of the desired compound probability divided by the sum of all the possible outcomes. Express the answer rounded if necessary to the nearest whole percentage, with percent symbol.

Question 10)

Consider the following context:

The nutrition and food science division of Farnsworth Pharmaceuticals is alpha testing an OTC diagnostic kit so that adults who self-report a certain food allergy can affirm their insensitivity to that food allergen. Preliminary clinical trials yielded that 25% of the volunteers who suspected they had food allergies were clinically identified as sensitive or intolerant to the food allergen; unfortunately and disappointingly, only 10% of those who tested positive in the clinic also tested positive with the kit. Furthermore, of those self-reporting adult volunteers who had a negative clinic outcome, 5% tested positive anyway with the kit. [COMMENTS & HINTS: Consider clinical sensitivity to the food allergen a positive result for testing purposes. If necessary, round off percentages to the nearest hundredth, expressing as #.##% or ##.##%.]

Given that the self-reporting adult did indeed test positive on the OTC diagnostic kit for food sensitivity, what is the chance the adult had a negative result on the clinical diagnosis in the first place???? HINT: Create a ratio of the desired compound probability divided by the sum of all the possible outcomes. Express the answer rounded to the nearest whole percentage, with percent symbol.

Question 11)

Consider the following context:

The nutrition and food science division of Farnsworth Pharmaceuticals is alpha testing an OTC diagnostic kit so that adults who self-report a certain food allergy can affirm their insensitivity to that food allergen. Preliminary clinical trials yielded that 25% of the volunteers who suspected they had food allergies were clinically identified as sensitive or intolerant to the food allergen; unfortunately and disappointingly, only 10% of those who tested positive in the clinic also tested positive with the kit. Furthermore, of those self-reporting adult volunteers who had a negative clinic outcome, 5% tested positive anyway with the kit. [COMMENTS & HINTS: Consider clinical sensitivity to the food allergen a positive result for testing purposes. If necessary, round off percentages to the nearest hundredth, expressing as #.##% or ##.##%.]

What is the not-so-surprising sum of the previous two conditional probability problems????? HINT: These probabilities can be referred to as posterior probabilities, since these ratios describe the chance of the condition, given the consequence. Express the answer as a whole percentage, with percent symbol.

Question 12)

Consider the following context:

The nutrition and food science division of Farnsworth Pharmaceuticals is alpha testing an OTC diagnostic kit so that adults who self-report a certain food allergy can affirm their insensitivity to that food allergen. Preliminary clinical trials yielded that 25% of the volunteers who suspected they had food allergies were clinically identified as sensitive or intolerant to the food allergen; unfortunately and disappointingly, only 10% of those who tested positive in the clinic also tested positive with the kit. Furthermore, of those self-reporting adult volunteers who had a negative clinic outcome, 5% tested positive anyway with the kit. [COMMENTS & HINTS: Consider clinical sensitivity to the food allergen a positive result for testing purposes. If necessary, round off percentages to the nearest hundredth, expressing as #.##% or ##.##%.]

Given that the self-reporting adult did indeed test tolerant or insensitive to the food allergen (i.e., a negative outcome) on the OTC diagnostic kit, what is the chance the adult indicated as intolerant or sensitive (i.e., a positive outcome) to the food allergen on the clinical diagnosis in the first place??????

HINT: Create a ratio of the desired compound probability divided by the sum of all the possible outcomes. Express the answer rounded to the nearest whole percentage, with percent symbol.

Question 13)

Consider the following context:

The nutrition and food science division of Farnsworth Pharmaceuticals is alpha testing an OTC diagnostic kit so that adults who self-report a certain food allergy can affirm their insensitivity to that food allergen. Preliminary clinical trials yielded that 25% of the volunteers who suspected they had food allergies were clinically identified as sensitive or intolerant to the food allergen; unfortunately and disappointingly, only 10% of those who tested positive in the clinic also tested positive with the kit. Furthermore, of those self-reporting adult volunteers who had a negative clinic outcome, 5% tested positive anyway with the kit. [COMMENTS & HINTS: Consider clinical sensitivity to the food allergen a positive result for testing purposes. If necessary, round off percentages to the nearest hundredth, expressing as #.##% or ##.##%.]

Given that the self-reporting adult did indeed tested tolerant or insensitive (i.e., a negative outcome) to the food allergen based on the OTC diagnostic kit, what is the chance the adult indicated as tolerant or insensitive (i.e., a negative outcome) to the food allergen on the clinical diagnosis in the first place????

HINT: Create a ratio of the desired compound probability divided by the sum of all the possible outcomes. Express the answer rounded to the nearest whole percentage, with percent symbol.

Question 14)

Consider the following context:

The nutrition and food science division of Farnsworth Pharmaceuticals is alpha testing an OTC diagnostic kit so that adults who self-report a certain food allergy can affirm their insensitivity to that food allergen. Preliminary clinical trials yielded that 25% of the volunteers who suspected they had food allergies were clinically identified as sensitive or intolerant to the food allergen; unfortunately and disappointingly, only 10% of those who tested positive in the clinic also tested positive with the kit. Furthermore, of those self-reporting adult volunteers who had a negative clinic outcome, 5% tested positive anyway with the kit. [COMMENTS & HINTS: Consider clinical sensitivity to the food allergen a positive result for testing purposes. If necessary, round off percentages to the nearest hundredth, expressing as #.##% or ##.##%.]

What is the not-so-surprising sum of the previous two conditional probability problems???? HINT: These probabilities can be referred to as posterior probabilities, since these ratios describe the chance of the condition, given the consequence. Express the answer as a whole percentage, with percent symbol.

Question 15)

Consider the following context:

The nutrition and food science division of Farnsworth Pharmaceuticals is alpha testing an OTC diagnostic kit so that adults who self-report a certain food allergy can affirm their insensitivity to that food allergen. Preliminary clinical trials yielded that 25% of the volunteers who suspected they had food allergies were clinically identified as sensitive or intolerant to the food allergen; unfortunately and disappointingly, only 10% of those who tested positive in the clinic also tested positive with the kit. Furthermore, of those self-reporting adult volunteers who had a negative clinic outcome, 5% tested positive anyway with the kit. [COMMENTS & HINTS: Consider clinical sensitivity to the food allergen a positive result for testing purposes. If necessary, round off percentages to the nearest hundredth, expressing as #.##% or ##.##%.]

As a final review, answer the following question:

What is the chance a volunteer had food allergies according to clinical results, given that the volunteer had a negative test result from the OTC diagnostic kit?????

HINT: These probabilities can be referred to as posterior probabilities, since these ratios describe the chance of the condition, given the consequence. Express the answer as a whole percentage, with percent symbol.

Question 16)

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