1. Suppose you are trying to estimate the average miles per gallon for a new brand of car. You take a random sample of 40 cars, and, for this sample, the average miles per gallon is 32 and the standard deviation for this sample is 2.2. Answer the following questions in a Word document: •What is the population mean you are trying to estimate?
What value is the point estimate for the population mean? Describe, in your own words, what the point estimate represents.
Suppose you would like to construct a 90 percent confidence interval. What would be the margin of error for this interval?
What would be the lower and upper limits for this 90 percent confidence interval?
Now that you have constructed the interval, define its meaning in your own words.
2. You are trying to estimate the average amount that movie theaters charge for a large bucket of popcorn at theaters in the United States. Now, of course, you can't call up every single theater in the U.S. and ask them how much they charge for a large bucket of popcorn because that would take too much time. All you want is a good estimate, so you will create a confidence interval.
You randomly sample 50 theaters in the United States. You ask those theaters how much they charge for a large popcorn, and you get a sample mean of $6. Then, you create a confidence interval using this data with the lower limit at $4.50 and the upper limit at $7.50.
Explain why a confidence interval would or would not be appropriate in this example.
In your own words, interpret the meaning of this confidence interval.
If you only had a sample of 10 theaters instead of 50, would a confidence interval still be appropriate? Why or why not?
Describe a scenario where you could use the confidence interval in your daily life.
3. Suppose a car dealer tells you that specific cars get 35 miles per gallon on average. You would be able to use hypothesis tests to test whether or not this value might be correct. Why would you use a hypothesis test instead of a confidence interval in this situation?
A car dealer states that a new brand of car gets 35 miles per gallon on average. Suppose a consumer group claims that these cars get less than 35 miles per gallon. Set up the null and alternative hypotheses for this example.
The average daily rainfall in a jungle in South America was four inches back in 2000. Suppose a scientist thinks the average rainfall is different now. Set up the null and alternative hypotheses for this example.
4. A car dealer states that a new brand of car gets 35 miles per gallon on average. Suppose a consumer group claims that these cars get less than 35 miles per gallon. A sample of 40 cars is selected, and the sample mean for the 40 cars is 33 miles per gallon while the sample standard deviation is 3.8. •Have the assumptions for this test been met? Why or why not?
State the null and alternative hypothesis for this test.
Calculate the test statistic for this test. Explain what this test statistic represents.
Use technology to calculate the p-value for this test. Explain what this p-value represents.
State the conclusion for this test at the 0.05 level of significance. Do you think the car dealer is telling the truth? Why or why not?
The solution gives detailed steps on answering a set of short answer questions on the topic of hypothesis testing: null and alternative hypothesis, test statistic, p-value and assumptions.
Hypothesis Testing: Null and Alternate Hypotheses, P Value
Four Hypothesis Testing problems
Problem 10.17: M&M/Mars claims that at least 20% of the M&M's in each package are the new blue color. Set up the null and alternate hypotheses to test this claim.
Problem 10.18: The computer center at a university claims that the average amount of time that students spend on-line has increased from last year's average of 1 hour per day. Set up the null and alternative hypotheses to test this claim.
Problem 10.20: Recent medical research indicates that skin cancer patients who receive a new medication for skin cancer live longer than those who do not. The average length of life prior to the development of this medication was 18 months. The medical community wishes to test the claim made by the developers of this drug. A sample of 35 patients who received the medication lived an average of 21 months. The standard deviation is 5 months.
a) Set up the null and the alternative hypotheses to test if average length of life has increased from 18 months.
b) Test your hypotheses using a = 0.05.
c) Find the p value.
d) Based on the p value, what can you conclude about the average length of life for patients who receive the vaccine?
Problem 10.22: A manufacturer of top-of-line tennis rackets claims that its Smack Em tennis racket will change a player's game. A tennis pro currently serves the ball at an average speed of 115 mph with a standard deviation of 2.5 mph. The speeds are normally distributed. The tennis pro decides to test the company's claim and records the speed of his serve for 15 balls using the Smack Em racket. The data are shown in the following table: Speed
a) Set up the null and the alternative hypotheses to test if the average service speed has increased using the new racket.
b) Test your hypotheses using a = 0.05
c) Find the p value.
d) Based on the p value, should the tennis pro invest in the new racket?