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Vlad [161]
3 years ago
7

Two production lines are used to pack sugar into 5 kg bags. Line 1 produces twice as many bags as does line 2. One percent of ba

gs from line 1 are defective in that they fail to meet a purity specification, while 3% of the bags from line 2 are defective. A bag is randomly chosen for inspection. ( 2 points, each part is 0.5 points) b. What is the probability that is defective
Mathematics
2 answers:
enyata [817]3 years ago
7 0

Answer:

P(Bag is Defective) = 0.0167

Step-by-step explanation:

Line 1 produces twice as many bags as line 2. Let x be the number of bags produced by line 2.

No. of bags produced by line 2 = x

No. of bags produced by line 1 = 2x

Probability that the bag has been produced by line 1 can be written as:

P(Line 1) = No. of bags produced by line 1/Total no. of bags

             = 2x/(x+2x)

             = 2x/3x

P(Line 1) = 2/3. Similarly,

P(Line 2) = x/3x

P(Line 2) = 1/3

1% bags produced by line 1 are defective so the probability of line 1 producing a defective bag is:

P(Defective|Line 1) = 0.01

3% of bags from line 2 are defective, so:

P(Defective|Line 2) = 0.03

b. The probability that the chosen bag is defective can be calculated through the conditional probability formula:

P(A|B) = P(A∩B)/P(B)

<u>P(A∩B) = P(A|B)*P(B)</u>

The chosen defective bag can be either from line 1 or from line 2. So, the probability that the chosen bag is defective is:

P(Bag is Defective) = P(Defective and from Line 1) + P(Defective and from Line 2)

                                = P(D∩Line 1) + P(D∩Line 2)

                                = P(Defective|Line 1)*P(Line 1) + P(Defective|Line 2)*P(Line 2)

                                = (0.01)*(2/3) + (0.03)(1/3)

P(Bag is Defective) = 0.0167

grigory [225]3 years ago
3 0

Answer:

P(z) = 5/300 = 0.0167 or 1.67%

The probability that a bag randomly chosen for inspection is defective is 0.0167

Step-by-step explanation:

Let x, y and z represent the number of bags of sugar produced by line 1, line 2 and both respectively.

z = x + y ....1

Line 1 produces twice as many bags as does line 2

x = 2y .......2

Substituting into equation 1

z = 2y + y

z = 3y  .........3

One percent of bags from line 1 are defective

N(x) = 0.01x= 0.01(2y)           (x = 2y)

N(x)  = 0.02y          

3% of the bags from line 2 are defective.

N(y) = 0.03y

The total number of defective bags from both lines

N(z) = 0.02y + 0.03y = 0.05y .....4

The probability that a bag randomly chosen for inspection is defective;

P(z) = N(z)/z  .......5

Substituting equation 3 and 4

P(z) = 0.05y/3y

P(z) = 5/300 = 0.0167 or 1.67%

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Answer:

Null hypothesis: p= 0.0147

Alternative hypothesis: p\neq 0.0147

A type of error II for this case would be FAIL to reject the null hypothesis that the population proportion is equal to 0.0147 when actually the alternative hypothesis is true (the true proportion is different from 0.0147).

Step-by-step explanation:

Previous concepts

A hypothesis is defined as "a speculation or theory based on insufficient evidence that lends itself to further testing and experimentation. With further testing, a hypothesis can usually be proven true or false".  

The null hypothesis is defined as "a hypothesis that says there is no statistical significance between the two variables in the hypothesis. It is the hypothesis that the researcher is trying to disprove".  

The alternative hypothesis is "just the inverse, or opposite, of the null hypothesis. It is the hypothesis that researcher is trying to prove".  

Type I error, also known as a “false positive” is the error of rejecting a null  hypothesis when it is actually true. Can be interpreted as the error of no reject an  alternative hypothesis when the results can be  attributed not to the reality.  

Type II error, also known as a "false negative" is the error of not rejecting a null  hypothesis when the alternative hypothesis is the true. Can be interpreted as the error of failing to accept an alternative hypothesis when we don't have enough statistical power.  

Solution to the problem

On this case we want to test if the proportion of children diagnosed with Autism Spectrum Disorder (ASD) is different from 0.0147, so the system of hypothesis would be:

Null hypothesis: p= 0.0147

Alternative hypothesis: p\neq 0.0147

A type of error II for this case would be FAIL to reject the null hypothesis that the population proportion is equal to 0.0147 when actually the alternative hypothesis is true (the true proportion is different from 0.0147).

7 0
3 years ago
Read 2 more answers
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