1. You are quality manager for an LED light bulb manufacturer. If your manufacturing process is running well, the failure rate for your light bulbs should be 0.05 (5%) as they come out of the production process. You want to design a sampling scheme to determine whether the process is under control. To inform you in this decision, you need to calculate a few probabilities based on a binomial distribution:
(a) If you sample 8 light bulbs, calculate the probability that one of them will be defective
(b) If you sample 8 light bulbs, calculate the probability that none of them will be defective
(c) If you sample 100 light bulbs, calculate the probability that more than 10 (10%) will be defective using the normal approximation to the binomial distribution.
(d) Suppose you sample 8 light bulbs and find that 2 of them are defective. If you want the process to shut down when the probability of failure has not truly changed less than 5% of the time, should you conclude something is wrong with the process and shut it down?
(Hint: Use the results from previous parts of the problem)
2. Find the following probabilities:
(a) Assuming Z is a random variable that follows the standard normal distribution with a mean of 0 and a standard deviation of 1, find P(Z > -0.47). Draw a standard normal distribution curve that shows how you found this probability.
(b) If X is a normally distributed random variable with a mean of 145 and a standard deviation of 10, find the probability that X is greater than 150.
(c) If µ = 30 and the sample size is 64 with σ = 8, are we justified in using the normal approximation to find probabilities associated with this sampling distribution? Justify with a theorem we used in class. State the results of the theorem.
(d) Using the information in part (c), find P( > 25).
3. Solve for the following:
(a) Given a normal population with µ = 25 and σ = 5, find the probability that an assumed value of the variable will be less than 30.
(b) The true weights of ten-pound sacks of potatoes processed at a certain packaging house have a normal distribution with mean 10 lbs and standard deviation of 1 lbs. What is the probability that a sack purchased at the grocery store will weigh at least 8 lbs.?
(c) Given a population of values for which the mean is 100 and the standard deviation is 15, find the probability that an assumed value will be within one standard deviation of the mean. Assume a normal distribution.
(d) A population of Australian Koala bears has a mean height of 20 inches and a standard deviation of 4 inches. You plan to choose a sample of 64 bears at random. What is the probability of a sample mean being above 21 inches?
4. In a massive attempt to compete with General Electric, the Acme Light Bulb Company issued a new line of bulb. Acme took 100 bulbs from their new line which had an established standard deviation of 140 hours. The mean measure lifetime was 1,280 hours.
(a) Calculate a 90% confidence interval for the mean lifetime of Acme's bulbs.
(b) Calculate a 95% confidence interval for the mean lifetime of Acme's bulbs.
(c) Calculate a 99% confidence interval for the mean lifetime of Acme's bulbs.
(d) What happens to the width of confidence intervals as the confidence level increases? Explain why this occurs.