Hello, I need help solving these problems. Thanks a lot. 1. A manufacturing defe

Question

Hello, I need help solving these problems. Thanks a lot.

1. A manufacturing defect occurs on average at a rate of 1.2 defects per 100 ft. spool of wire. Assuming that the Poisson model applies, what is the probability of finding at least one defect in a lot that contains 4 spools?

2.

The weight of a molded part is assumed to be normally distributed with a mean of 15.0 grams and a standard deviation of 1.1 grams.

(a)       What is the probability that a randomly selected part weighs less than 14.0 grams?

(b)       What is the 5th percentile for the distribution of parts (i.e., 5% of parts weigh less than and 95% weigh more than what value)?

(c)        How much does the mean need to be increased in order for there to be at most a 5% chance that an individual part weighs less than 14.0 grams assuming that the variation is unchanged (standard deviation remains at 1.1 grams)?

(d)       How much of a reduction in variation is necessary in order for there to be at most a 5% chance that an individual part weighs less than 14.0 grams assuming that the mean remains at 15.0 grams?

9.Congenital heart disease affects 1 out of every 100 live born infants in the United States. A Minnesota researcher is going to randomly sample 1,000 birth records in one Minnesota county and compare the county rate to the national rate. Let X represent the number of infants diagnosed with congenital heart disease in the sample from the Minnesota county.

(a)What assumptions are required for X to be considered a binomial random variable?

(b)Assuming that the binomial model applies and that the national rate applies to the Minnesota county, provide the sampling distribution for X. In other words, graph P(X) vs. X.

(c)What is the mean and standard deviation for X? Is it reasonable for a normal distribution to approximate the distribution of X?

(d)The sample from the Minnesota county shows that there were 15 infants diagnosed with congenital heart disease. Given the national rate, do you consider the observation of 15 infants affected out of 1,000 births to be statistically unusually high? State why or why not.

3. A manufacturing company has discovered a problem that it believes has been contained within the manufacturing plant. It wants to conduct an audit of finished goods inventory at its distribution center to check for the defect and to ensure that the problem was contained in manufacturing. There are thousands of products at the distribution site that can be inspected.

(a)       Assuming that no defects are found, what sample size is needed to generate evidence that the defect rate in finished goods is less than 1% with 95% confidence?

(b)       The company conducts an inspection of n=250 randomly selected units from finished goods inventory and finds that 1 unit contains the defect. Give a point estimate for the defect rate and construct a 95% upper confidence bound for the defect rate in the entire finished goods population. Is there sufficient statistical evidence to show that the defect rate in the population is below 1% at the desired level of confidence?

Explanation / Answer

Ans:

1)mean=1.2*4=4.8 defects per 4 spools

P(x>=1)=1-P(x=0)=1-e-4.8*(4.80/0!)=1-e-4.8=1-0.0082=0.9918

2)mean=15

standard deviation=1.1

a)

z=(14-15)/1.1=-1/1.1=-0.91

P(z<-0.91)=0.1814

b)P(Z<=z)=0.05

z=-1.645

x=15-1.645*1.1=15-1.81=13.19

c)

z=(14-mean)/1.1

Now,

P(Z<=z)=0.05

z=-1.645

So,

-1.645=(14-mean)/1.1

mean=14+1.81=15.81

d)

-1.645=(14-15)/std dev.

std dev=1/1.645=0.61

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