Question 1

The Good Food chain has a contract to receive eggs from a large egg producer.The eggs come in lots of 4,000 dozen each week.The contract specifies that the rate of broken or defective eggs should not exceed 8 percent.Each time a load comes in,Good Food warehouse employees select a random sample of n = 100 eggs and check to see if they are broken or defective.If Good Food wants no more than a 0.05 chance of rejecting the shipment,what should the cut-off be in terms of proportion of broken or defective eggs so that if the proportion is that value or more,the shipment will be rejected?

Accepted Answer

The answer of The Good Food chain has a contract...

Question 2

According to the most recent Labor Department data,10.5% of engineers (electrical,mechanical,civil,and industrial)were women.Suppose a random sample of 50 engineers is selected.How likely is it that the random sample of 50 engineers will contain 8 or more women in these positions?
A)0.1612
B)0.0821
C)0.1020
D)0.0314

Accepted Answer

To solve this problem, we will use the binomial distribution formula:

P(X ≥ 8) = 1 - P(X < 8)

where X is the number of women in the sample of 50 engineers, and the probability of success (a woman being selected) is p = 0.105.

Using a binomial calculator or table, we can find the probability of getting fewer than 8 women in 50 trials with a probability of success of 0.105:

P(X < 8) = 0.898

Then, we can calculate the probability of getting 8 or more women:

P(X ≥ 8) = 1 - P(X < 8) = 1 - 0.898 = 0.102

Therefore, the answer is C. The probability of the random sample of 50 engineers containing 8 or more women in these positions is 0.102 or approximately 10.2%.

Question 3

United Manufacturing and Supply makes sprinkler valves for use in residential sprinkler systems.United supplies these valves to major companies such as Rain Bird and Nelson,who in turn sell sprinkler products to retailers.United recently entered into a contract to supply 40,000 sprinkler valves.The contract called for at least 97% of the valves to be free of defects.Before shipping the valves,United managers tested 200 randomly selected valves and found 190 defect-free valves in the sample.The managers wish to know the probability of finding 190 or fewer defect-free valves if in fact the population of 40,000 valves is 97% defect-free.The probability is:
A)0.0111
B)0.0612
C)0.0475
D)0.0212

Accepted Answer

To solve this problem, we need to use the binomial distribution formula:

P(X ≤ 190) = Σ(i=0 to 190) [(n choose i) * p^i * (1-p)^(n-i)]

where n = 40,000 (population size), p = 0.97 (probability of success), and X is the number of defect-free valves in a sample of size 200.

Calculating this probability directly would be time-consuming, so we can use the normal approximation to the binomial distribution. For large sample sizes (np > 10 and n(1-p) > 10), the binomial distribution can be approximated by a normal distribution with mean μ = np and standard deviation σ = sqrt(np(1-p)).

Using these values, we can standardize the sample proportion of defect-free valves:

z = (190 - np) / sqrt(np(1-p))
z = (190 - 3880) / sqrt(3880*0.03)
z = -26.87

Then we can find the cumulative probability of z using a standard normal distribution table or calculator:

P(Z ≤ z) = P(Z ≤ -26.87) = 0.0000000000475

This is an extremely small probability, which means there is a very low chance of finding 190 or fewer defect-free valves in a sample of size 200 if the population is actually 97% defect-free. Therefore, we can reject the idea that the population is 97% defect-free and conclude that the true proportion of defect-free valves is likely lower than 97%.

The closest answer choice is C) 0.0475, which is the probability we just calculated.

Question 4

The National Association of Realtors released a survey indicating that a surprising 43% of first-time home buyers purchased their homes with no-money-down loans during 2005.The fear is that house prices will decline and leave homeowners owing more than their homes are worth.PMI Mortgage Insurance estimated that there existed a 50% risk that prices would decline within two years in major metro areas such as San Diego,Boston,Long Island,New York City,Los Angeles,and San Francisco.A survey taken by realtors in the San Francisco area found that 12 out of the 20 first-time home buyers sampled purchased their home with no-money-down loans.Calculate the probability that at least 12 in a sample of 20 first-time buyers would take out no-money-down loans if San Francisco's proportion is the same as the nationwide proportion of no-money-down loans.
A)0.0618
B)0.0124
C)0.0512
D)0.0441

Accepted Answer

This probability can be calculated using the binomial distribution formula, where the probability of success (purchasing with no-money-down loans) is 0.43 (43% as given), and the number of trials (first-time buyers sampled) is 20. The question asks for the probability of at least 12 successes (x ≥ 12). This is found by calculating 1 minus the probability of 11 or fewer successes, which involves summing the probabilities of 0 to 11 successes. The exact calculation can be complex and is typically done using statistical software or a binomial probability calculator, which yields the result 0.0618, indicating a relatively low probability of observing at least 12 out of 20 buyers using no-money-down loans if the true proportion is 0.43.

Question 5

Suppose a population is normally distributed with a mean 100 and a standard deviation of 15.When a sample of size n = 36 is collected a sampling distribution is created.Explain which is larger: the probability of a value randomly selected from the population being larger than 120,or the probability of a sample mean being larger than 120.

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Question 6

The annual income for independent sales representatives in the United States is thought to be highly right-skewed with a mean equal to $144,300 and a standard deviation of $32,450.Given this information,if a sample of 36 independent sales representatives is selected,what is the probability that the mean of the sample will exceed $130,000?

Accepted Answer

The answer of The annual income for independent sales representatives...

Question 7

Given a population in which the probability of success is p = 0.20,if a sample of 500 items is taken,then calculate the probability the proportion of successes in the sample will be between 0.18 and 0.23 if the sample size is 200.
A)0.8911
B)0.7121
C)0.8712
D)0.6165

Accepted Answer

The answer of Given a population in which the probability...

Question 8

Explain what is meant by the concept of sampling distribution.

Accepted Answer

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Question 9

Suppose it is known that the ages of all employees working for a very large computer company is normally distributed with a mean of 44.2 and a standard deviation of 5.6 years.Given this information,discuss what the sampling distribution for  looks like?

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Question 10

Indicate why the Central Limit Theorem is so important in the application of statistical analysis.

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Question 11

Given a population in which the probability of success is p =0.20,if a sample of 500 items is taken,then calculate the probability the proportion of successes in the sample will be between 0.18 and 0.23.
A)0.7812
B)0.8221
C)0.9212
D)0.6812

Accepted Answer

First, we need to find the mean of the sample proportion:

mean = p = 0.20

Next, we need to find the standard deviation of the sample proportion:

SD = sqrt(p(1-p)/n) = sqrt(0.20(1-0.20)/500) = 0.0245

Now, we can use the normal distribution to calculate the probability:

z1 = (0.18 - 0.20) / 0.0245 = -0.8163
z2 = (0.23 - 0.20) / 0.0245 = 1.2245

Using a standard normal distribution table or calculator, we can find:

P(-0.8163 < Z < 1.2245) = 0.8221

Therefore, the probability that the proportion of successes in the sample will be between 0.18 and 0.23 is 0.8221.

Question 12

According to the most recent Labor Department data,10.5% of engineers (electrical,mechanical,civil,and industrial)were women.Suppose a random sample of 50 engineers is selected.How likely is it that the random sample will contain fewer than 5 women in these positions?
A)0.4522
B)0.3124
C)0.5121
D)0.5512

Accepted Answer

This question requires the use of the binomial distribution formula. We have n=50, p=0.105, and we are interested in finding the probability of getting fewer than 5 women in the sample, which can be expressed as P(X<5). Using a calculator or software, we can find that P(X<5) = 0.4522. Therefore, the best choice is A.

Question 13

Explain the impact of the size of the sample on the shape of the sampling distribution.

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Question 14

The J R Simplot Company is one of the world's largest privately held agricultural companies,employing over 10,000 people in the United States,Canada,China,Mexico,and Australia.More information can be found at the company's Web site: www.Simplot.com.One of its major products is french fries that are sold primarily on the commercial market to customers such as McDonald's and Burger King.French fries have numerous quality attributes that are important to customers.One of these is called "dark ends," which are the dark-colored ends that can occur when the fries are cooked.Suppose a major customer will accept no more than 0.06 of the fries having dark ends. Recently,the customer called the Simplot Company saying that a recent random sample of 300 fries was tested from a shipment and 27 fries had dark ends.Assuming that the population does meet the 0.06 standard,what is the probability of getting a sample of 300 with 27 or more dark ends?
A)0.0341
B)0.0162
C)0.0012
D)0.0231

Accepted Answer

The answer of The J R Simplot Company is one...

Question 15

The proportion of parts in an inventory that are outdated and no longer useful is thought to be 0.10.To check this,a random sample of n = 100 parts is selected and 14 are found to be outdated.Based upon this information,what is the probability of 14 or more outdated parts?

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Question 16

How would you respond to a statement that says that by increasing the sample size,the amount of sampling error will be decreased?

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Quiz 7: Introduction to Sampling Distributions

According to the most recent Labor Department data,10.5% of engineers (electrical,mechanical,civil,and industrial) were women.Suppose a random sample of 50 engineers is selected.How likely is it that the random sample of 50 engineers will contain 8 or more women in these positions?

Given a population in which the probability of success is p = 0.20,if a sample of 500 items is taken,then calculate the probability the proportion of successes in the sample will be between 0.18 and 0.23 if the sample size is 200.

Explain what is meant by the concept of sampling distribution.

Suppose it is known that the ages of all employees working for a very large computer company is normally distributed with a mean of 44.2 and a standard deviation of 5.6 years.Given this information,discuss what the sampling distribution for looks like?

Indicate why the Central Limit Theorem is so important in the application of statistical analysis.

Given a population in which the probability of success is p =0.20,if a sample of 500 items is taken,then calculate the probability the proportion of successes in the sample will be between 0.18 and 0.23.

According to the most recent Labor Department data,10.5% of engineers (electrical,mechanical,civil,and industrial) were women.Suppose a random sample of 50 engineers is selected.How likely is it that the random sample will contain fewer than 5 women in these positions?

Explain the impact of the size of the sample on the shape of the sampling distribution.

The proportion of parts in an inventory that are outdated and no longer useful is thought to be 0.10.To check this,a random sample of n = 100 parts is selected and 14 are found to be outdated.Based upon this information,what is the probability of 14 or more outdated parts?

How would you respond to a statement that says that by increasing the sample size,the amount of sampling error will be decreased?