1. Exponential
The time it takes a worker on an assembly line to complete a task is exponentially distributed with a mean of 8 minutes. Please answer the following questions:
a. What is the probability density function for the time it takes to complete the task?
b. What is the probability that it will take a worker less than 4 minutes to complete the task?
c. What is the probability that it will take a worker between 6 and 10 minutes to complete the task?
[Hint: Please see Chap006 - Slides 45-50 for Exponential probability distribution. Please also see page 285-287 in the textbook.]
2. Sampling Distribution
The average lifetime of a light bulb is 3,000 hours with a standard deviation of 696 hours. A simple random sample of 36 bulbs is taken. Please answer the following questions:
a. What are the expected value, standard deviation, and shape of the sampling distribution of ?
b. What is the probability that the average life in the sample will be between 2,884 and 3,116 hours?
c. What is the probability that the average life in the sample will be between 2,768 and 3,232 hours?
[Hint: Please see Chap007 - Slides 20 -35 for formula and example. Please also see page 310-316 in the textbook.]
3. Estimate Interval
The makers of a soft drink want to identify the average age of its consumers. A sample of 55 consumers was taken. The average age in the sample was 21 years with a sample standard deviation of 4 years. Please answer the following questions:
a. Construct a 95% confidence interval estimate for the mean of the consumers' age.
b. Suppose a sample of 71 was selected (with the same mean and the sample standard deviation). Construct a 95% confidence interval for the mean of the consumers' age.
[Hint: Please see Chap008 - Slides 24-29 for formula and example. Please also see page 343-349 in the textbook.]
4. Hypothesis Testing
Choo Choo Paper Company makes various types of paper products. One of their products is a 30 mils thick paper. In order to ensure that the thickness of the paper meets the 30 mils specification, random cuts of paper are selected and the thickness of each cut is measured. A sample of 256 cuts had a mean thickness of 30.3 mils with a standard deviation of 4 mils.
a. Compute the standard error of the mean.
b. At 95% confidence using the critical value approach, test to see if the mean thickness is significantly more than 30 mils.
Extra credit
5. A lathe is set to cut bars of steel into lengths of 6 centimeters. The lathe is considered to be in perfect adjustment if the average length of the bars it cuts is 6 centimeters. A sample of 121 bars is selected randomly and measured. It is determined that the average length of the bars in the sample is 6.08 centimeters with a standard deviation of 0.44 centimeters.
a. Formulate the hypotheses to determine whether or not the lathe is in perfect adjustment.
b. Compute the test statistic.
c. Using the p-value approach, what is your conclusion? Let α = .05.