Question 1

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 2

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 3

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 of 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 4

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 5

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. 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 aforementioned interval?

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Question 6

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 7

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 heights (in inches) are 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 8

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 9

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 10

An employee of the Bureau of Transportation Statistics has been given the task of estimating the proportion of on-time arrivals of a budget airline. A prior study has estimated this on-time arrival rate as 78.5%. What is the minimum number of arrivals this employee must include in the sample to ensure that the margin of error for a 95% confidence interval is no more than 0.05?
A) 270
B) 259
C) 260
D) 269

Accepted Answer

The formula for the sample size of a proportion is:n = (Z^2 * p * q) / e^2where:n is the sample sizeZ is the z-score for the desired confidence levelp is the estimated proportion of successesq is the estimated proportion of failures (1 - p)e is the margin of errorIn this case, we have:Z = 1.96 (for a 95% confidence level)p = 0.785q = 1 - p = 0.215e = 0.05Plugging these values into the formula, we get:n = (1.96^2 * 0.785 * 0.215) / 0.05^2= 260.06Therefore, the minimum number of arrivals this employee must include in the sample is 260.

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 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 13

The Retail Advertising and Marketing Association would like to estimate the average amount of money that a person spends for Mother's Day with 99% confidence interval and a margin of error within plus or minus $6. Assuming the standard deviation for spending on Mother's Day is $36, the required sample size is ________.
A) 210
B) 239
C) 284
D) 313

Accepted Answer

To calculate the required sample size for estimating the average amount of money spent on Mother's Day with a specific margin of error and confidence level, we use the formula for sample size in estimating a mean: $n = \left(\frac{Z\sigma}{E}\right)^2$, where $n$ is the sample size, $Z$ is the Z-score corresponding to the desired confidence level, $\sigma$ is the population standard deviation, and $E$ is the margin of error. For a 99% confidence interval, the Z-score is approximately 2.576. Given $\sigma = $36$ and $E = $6$, substituting these values into the formula gives $n = \left(\frac{2.576 \times 36}{6}\right)^2$, which calculates to approximately 239. Therefore, the required sample size is 239.

Question 14

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 15

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

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Question 16

The National Center for Education would like to estimate the proportion of students who defaulted on their student loans for the state of Arizona. The total sample size needed to construct a 95% confidence interval for the proportion of student loans in default with a margin of error equal to 4% is ________.
A) 336
B) 455
C) 416
D) 601

Accepted Answer

To calculate the sample size for a proportion with a given margin of error, we use the formula n = (Z^2 * p * (1-p)) / E^2. Assuming p = 0.5 for maximum variability, Z = 1.96 for 95% confidence, and E = 0.04, the calculation results in approximately 601.

Question 17

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 18

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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Question 19

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

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Question 20

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.

Accepted Answer

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Quiz 8: Interval Estimation

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

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