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Statistics: t-scores, z-scores, confidence intervals, etc.

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10 TOTAL QUESTIONS INVOLVING Z-SCORES, CONFIDENCE INTERVALS, AND PROBABILITY

1. Polychlorinated biphenyl (PCB) is among a group of organic pollutants found in a variety of products, such as coolants, insulating materials, and lubricants in electrical equipment. Disposal of items containing less than parts per million (ppm) PCB is generally not regulated. A certain kind of small capacitor contains PCB with a mean of ppm and a standard deviation of ppm. The Environmental Protection Agency takes a random sample of of these small capacitors, planning to regulate the disposal of such capacitors if the sample mean amount of PCB is ppm or more. Find the probability that the disposal of such capacitors will be regulated.

What is the probability that the sample mean will be 49.5 or more?

2. A study of college football games shows that the number of holding penalties assessed has a mean of 2.3 penalties per game and a standard deviation of 0.9 penalties per game. What is the probability that, for a sample of 40 college games to be played next week, the mean number of holding penalties will be 2 penalties per game or less? Carry your intermediate computations to at least four decimal places. Round your answer to at least three decimal places.

What is the probability that the sample mean will be 2 or less?

3. A soft drink company has recently received customer complaints about its one-liter-sized soft drink products. Customers have been claiming that the one-liter-sized products contain less than one liter of soft drink. The company has decided to investigate the problem. According to the company records, when there is no malfunctioning in the beverage dispensing unit, the bottles contain 1.01 liters of beverage on average, with a standard deviation of 0.14 liters. A sample of 70 bottles has been taken to be measured from the beverage dispensing lot. The mean amount of beverage in these 70 bottles was 0.982 liters. Find the probability of observing a sample mean of 0.982 liters or less in a sample of 70 bottles, if the beverage dispensing unit functions properly.

What is the probability that the sample mean will be 0.982 or less?

4. The mean salary offered to students who are graduating from Coastal State University this year is , with a standard deviation of . A random sample of Coastal State students graduating this year has been selected.

What is the probability that the mean salary offer for these students is or more?

6. You would like to estimate the mean price of milk (per gallon) in your city. You select a random sample of prices from different stores. The sample has a mean of dollars and a standard deviation of dollars. For each of the following sampling scenarios, determine which test statistic is appropriate to use when making inference statements about the population mean.

(In the table, refers to a variable having a standard normal distribution, and refers to a variable having a t distribution.)

7. The breaking strengths of cables produced by a certain manufacturer have a standard deviation of pounds. A random sample of newly manufactured cables has a mean breaking strength of pounds. Based on this sample, find a confidence interval for the true mean breaking strength of all cables produced by this manufacturer.

Then complete the table below.
Carry your intermediate computations to at least three decimal places. Round your answers to one decimal place.

8. Managers at an automobile manufacturing plant would like to estimate the mean completion time of an assembly line operation, . The managers plan to choose a random sample of completion times and estimate via the sample. Assuming that the standard deviation of the population of completion times is minutes, what is the minimum sample size needed for the managers to be confident that their estimate is within minutes of ? Carry your intermediate computations to at least three decimal places. Write your answer as a whole number (and make sure that it is the minimum whole number that satisfies the requirements).

What is the probability that the sample mean will be 25000 or more?

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For all of these problems, we're going to assume that the distributions are normal and use the z-distribution. This means we're going to use the formula

z = (x - mu)/sigma

where mu is the population mean and sigma is the population standard deviation. (Note: for sample means, you divide sigma by the square root of the sample size.)

We will generally compare our value of z to a z-distribution table (http://www.statsoft.com/textbook/distribution-tables/) or use a calculator (http://faculty.vassar.edu/lowry/ch6apx.html) to find the probability.

1. Polychlorinated biphenyl (PCB) is among a group of organic pollutants found in a variety of products, such as coolants, insulating materials, and lubricants in electrical equipment. Disposal of items containing less than parts per million (ppm) PCB is generally not regulated. A certain kind of small capacitor contains PCB with a mean of ppm and a standard deviation of ppm. The Environmental Protection Agency takes a random sample of of these small capacitors, planning to regulate the disposal of such capacitors if the sample mean amount of PCB is ppm or more. Find the probability that the disposal of such capacitors will be regulated.

What is the probability that the sample mean will be 49.5 or more?

n = 39
σ = 8
µ = 48.2
x = 49.5

z = 49.5 - 48.2
8/√39

z = 1.3/1.2810 = 1.0148

The probability (area under the z-distribution curve) to the right of z = 1.0148 is p = 0.1551.

ANSWER: The probability is 15.51%.

2. A study of college football games shows that the number of holding penalties assessed has a mean of 2.3 penalties per game and a standard deviation of 0.9 penalties per game. What is the probability that, for a sample of 40 college games to be played next week, the mean number of holding penalties will be 2 penalties per game or less? Carry your intermediate computations to at least four decimal places. Round your answer to at least three decimal places.

What is the probability that the sample mean will be 2 or less?

n = 40
σ = 0.9
µ = 2.3
x = 2

z ...

$2.19