Exercise 1
The amount of money spent by a customer at a discount store has a mean of $100 and a standard deviation of $30. What is the probability that a randomly selected group of 50 shoppers will spend a total of more than $5300?
Exercise 2
A company manufactures ball bearings for precision machines. The average diameter of a certain type of ball bearing should be 6.0 mm.
To check that the average diameter is correct, the company formulates a statistical test.
a. State your null and research hypotheses.
b. Suppose the manufacturer requires a 1% level of significance. Describe a type I error, its consequence, and its probability.
a. Discuss a type II error and its consequences.
Exercise 3
A study by Consumer Reports showed that 64% of supermarket shoppers believe supermarket brands to be as good as national name brands. To investigate whether this result applies to its own product, the manufacturer of a national name-brand ketchup asked a sample of shoppers whether they believed that supermarket ketchup was as good as the national brand ketchup.
a. Formulate the hypotheses that could be used to determine whether the percentage of supermarket shoppers who believe that the supermarket ketchup was as good as the national brand ketchup differed from 64%.
b. If a sample of 100 shoppers showed 52 stating that the supermarket brand was as good as the national brand, what is the p-value?
c. At α=0.05, what is your conclusion?
d. Should the national brand ketchup manufacturer be pleased with this conclusion? Explain.
Exercise 4
What price do farmers get for their watermelon crops? In the third week of July, a random sample of 40 farming regions gave a sample mean of per x = $6.88 per 100 pounds of watermelon. Assume that σ is known to be $1.92 per 100 pounds.
a. Find a 90% confidence interval for the population mean price (per 100 pounds) that farmers in this region get for their watermelon crop. What is the bound on the error?
b. Find the sample size necessary for a 90% confidence level with the bound on the error B=0.3 for the mean price per 100 pounds of watermelon.
c. A farm brings 15 tons of watermelon to market. Find a 90% confidence interval for the population mean cash value of this crop. What is the bound on the error? Hint: 1 ton is 2000 pounds.
Exercise 5
a. Layla computed a confidence interval for µ based on a sample of size 41. Since she did not know σ, she used s in her calculations. Layla used the normal distribution for the confidence interval instead of a Student's t distribution. Was her interval longer or shorter than one obtained by using an appropriate Student's distribution? Explain.
b. Layla was in a hurry when she computed a confidence interval for µ. Because σ was not known, she used a Student's t distribution. However, she accidentally used degrees of freedom n instead of n-1. Was her confidence interval longer or shorter than one found using the correct degrees of freedom n-1? Explain.
Exercise 6
The fan blades on commercial jet engines must be replaced when wear on these parts indicates too much variability to pass inspection. If a single fan blade broke during operation, it could severely endanger a flight. A large engine contains thousands of fan blades, and safety regulations require that variability measurements on the population of all blades not exceed σ2 = 0.18 mm2. An engine inspector took a random sample of 61 fan blades from an engine.
She measured each blade and found a sample variance of 0.27 mm2. Using a 1% level of significance, is the inspector justified in claiming that all the engine fan blades must be replaced?