Question 1

A string quartet in Maui wants to better understand their income stream,which primarily comes from playing at weddings.The cellist records the number of weddings they play in six successive months.Construct a 90% confidence interval from her results: 3,5,7,4,4,and 5.

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Question 2

A company manager thinks he is overpaying for fertilizer at $174 per ton.To find out,he collects fertilizer prices per ton from five distributors.Construct a 90% confidence interval from his results: 128,140,170,166,and 156.

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Question 3

A large university is interested in the outcome of a course standardization process.They have taken a random sample of 100 student grades,across all instructors.The grades represent the proportion of problems answered correctly on a midterm exam.The sample proportion correct was calculated as 0.78.
A)Construct a 90% confidence interval on the population proportion of correctly answered problems.
B)Construct a 95% confidence interval on the population proportion of correctly answered problems.

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Question 4

The personnel department of a large corporation wants to estimate the family dental expenses of its employees to determine the feasibility of providing a dental insurance plan.A random sample of six employees reveals the following family dental expenses: $180,$260,$60,$40,$100,and $80.It is known that dental expenses follow a normal distribution.Construct a 90% confidence interval for the average family dental expenses for all employees of this corporation.

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Question 5

A sample of holiday shoppers is taken randomly from a local mall.Forty-nine shoppers were selected and asked what their average spending on gifts would be during the entire holiday season.The point estimate of the population mean was calculated as $550 and the sample standard deviation was calculated as $92.
A)Construct a 95% confidence interval of the population mean spending.
B)Explain how the central limit theorem is used in constructing this interval.

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Question 6

According to a report in USA Today (February 1,2012),more and more parents are helping their young adult children buy homes.You would like to construct a 90% confidence interval of the proportion of all young adults in Louisville,Kentucky,who got help from their parents in buying a home.How large a sample should you take so that the margin of error is no more than 0.06?
A)n = 114
B)n = 188
C)n = 267
D)Cannot be determined

Accepted Answer

We need to find the sample size to construct a 90% confidence interval with a margin of error no more than 0.06.

Using the formula for margin of error:

Margin of error = z*sqrt(p*(1-p)/n)

We need to find n.

The critical value for a 90% confidence interval is 1.645.

We don't know the value of p, so we use 0.5 as a conservative estimate to get the largest sample size:

0.06 = 1.645*sqrt(0.5*(1-0.5)/n)

Solving for n:

n = (1.645/0.06)^2 * 0.5*(1-0.5)

n = 187.8

Rounding up to the nearest whole number, we get

n = 188

Therefore, the best choice is B.

Question 7

The snowfall (in inches)during the month of January in a particular geographic region is normally distributed with a standard deviation of 16.75.In the last 12 years,the sample average of snowfall is computed as 122.50.
A)Construct a 90% confidence interval of the average snowfall in this region.
B)Construct a 95% confidence interval of the average snowfall in this region.
C)Comment on the width of the preceding confidence intervals.

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Question 8

In order to estimate the mean earnings forecast for a large company,an investor looked up earnings forecasts from six financial analysts.The six forecasts he found were 200,220,300,185,210,and 210 (in millions).Suppose the investor knows the population standard deviation is 25 million.Calculate a 95% confidence interval of the population mean earnings forecast.

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Question 9

The height of high school basketball players is known to be normally distributed with a standard deviation of 1.75 inches.In a random sample of eight high school basketball players,the height (in inches)is recorded as 75,82,68,74,78,70,77,and 76.Construct a 95% confidence interval on the average height of all high school basketball players.

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Question 10

Pure-bred dogs competing in shows must often meet specific height criteria to be within the breed standard.A new breed of dog,the pug-cock-a-poo,has been approved by a kennel club.In order to establish the breed standard,the organization of breeders has taken the height from a sample of 25 pug-cock-a-poos and found the sample mean to equal 14 inches.The variance of this sample is 2.25.
A)Construct a 90% confidence interval for the population mean height of the pug-cock-a-poos.
B)What assumption is necessary for constructing the above interval?

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Question 11

An environmentalist is measuring the number of spotted leopard lizards in central California per acre.The environmentalist surveys six acres and finds signs of 24,26,30,27,28,and 27 lizards per acre.Construct a 95% confidence interval for the average number of leopard lizards per acre.

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Question 12

A researcher in campaign finance law wants to estimate the proportion of elementary,middle,and high school teachers who contributed to a candidate during a recent election cycle.Given that no prior estimate of the population proportion is available,what is the minimum sample size such that the margin of error is no more than 0.03 for a 95% confidence interval?
A)751.67
B)752
C)1067.11
D)1068

Accepted Answer

The formula for minimum sample size needed to estimate a population proportion with a given margin of error is: 
n = (z^2 * p * q) / (E^2) 
Where: 
z = the critical value from the standard normal distribution corresponding to the desired level of confidence (1.96 for 95% confidence) 
p = the estimate of the population proportion (since there is no prior estimate, we use 0.5 for maximum variability) 
q = 1 - p 
E = the desired margin of error 
Plugging in the values, we get: 
n = (1.96^2 * 0.5 * 0.5) / (0.03^2) 
n = 1067.11
Therefore, the minimum sample size needed is 1068. Answer choice D is the closest option to this value.

Question 13

A sample of the weights of 35 babies is taken from the local hospital maternity ward.The point estimate for the mean weight of the babies is 110.2 ounces with a sample standard deviation of 23.4 ounces.Construct a 90% confidence interval for the population mean baby weight at this hospital.Interpret this interval.

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Question 14

A large university is interested in the outcome of a course standardization process.They have learned that 150 students of the total 1,500 students failed to pass the course in the current semester.Construct a 99% confidence interval on the population proportion of students who failed to pass the course.Was the normality condition met for the validity of the confidence interval formula?

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Question 15

A medical engineering company creates x-ray machines.The machines the company sold in 1995 were expected to last six years before breaking.To test how long the machines actually lasted,the company took a simple random sample of six machines.The company got the following results (in years)for how long the x-ray machines lasted: 8,6,7,9,5,and 7.Assume the distribution of the longevity of x-ray machines is normally distributed.Construct a 98% confidence interval for the average longevity of x-ray machines.

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Question 16

Bobby does not want to be late to work again.He drives to work every morning from Oakland to San Francisco and crossing the Bay Bridge seems to take forever.He times himself crossing the bridge for seven consecutive mornings,with the resulting times: 22.34,27.54,15.26,13.56,18.57,21.22,and 19.08 minutes.Construct a 99% confidence interval from those times.

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Question 17

A large city recently surveyed 1,354 street lights,finding that 4.2% of them had burned out.Construct a 99% confidence interval for the proportion of the city's street lights that are burned out.

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Question 18

A job candidate with an offer from a prominent investment bank wanted to estimate how many hours she would have to work per week during her first year at the bank.She took a sample of six first-year analysts,asking how many hours they worked in the last week.Construct a 95% confidence interval with her results: 64,82,74,73,78,and 87 hours.

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Question 19

A politician wants to estimate the percentage of people who like his new slogan.Given that no prior estimate of the population proportion is available,what is the minimum sample size such that the margin of error is no more than 0.08 for a 95% confidence interval?
A)105.70
B)106
C)150.06
D)151

Accepted Answer

The formula for calculating the minimum sample size is:
n = (Z^2 * p * q) / E^2
where:
Z = the Z-score corresponding to the desired level of confidence (1.96 for 95% confidence)
p = an estimate of the population proportion (since no prior estimate is available, we assume p = 0.5)
q = 1 - p
E = the maximum margin of error (0.08 in this case)

Substituting the values given,
n = (1.96^2 * 0.5 * 0.5) / 0.08^2
n = 151.06

Therefore, the minimum sample size needed is 151. Answer is D.

Question 20

Each portion of the SAT exam is designed to be normally distributed such that it has a population standard deviation of 100 and a mean of 500.However,the mean has changed over the years as less selective schools began requiring the SAT,and because students later began to prepare more specifically for the exam.Construct a 90% confidence interval for the population mean from the following eight scores from the math portion,using the population standard deviation of 100: 450,660,760,540,420,430,640,and 580.

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Quiz 8: Interval Estimation

A string quartet in Maui wants to better understand their income stream,which primarily comes from playing at weddings.The cellist records the number of weddings they play in six successive months.Construct a 90% confidence interval from her results: 3,5,7,4,4,and 5.

A company manager thinks he is overpaying for fertilizer at $174 per ton.To find out,he collects fertilizer prices per ton from five distributors.Construct a 90% confidence interval from his results: 128,140,170,166,and 156.

An environmentalist is measuring the number of spotted leopard lizards in central California per acre.The environmentalist surveys six acres and finds signs of 24,26,30,27,28,and 27 lizards per acre.Construct a 95% confidence interval for the average number of leopard lizards per acre.

A large city recently surveyed 1,354 street lights,finding that 4.2% of them had burned out.Construct a 99% confidence interval for the proportion of the city's street lights that are burned out.

A politician wants to estimate the percentage of people who like his new slogan.Given that no prior estimate of the population proportion is available,what is the minimum sample size such that the margin of error is no more than 0.08 for a 95% confidence interval?