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4.0
Scenario
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.
Instructions
1. Complete the following on the Discussion Board.
• 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.
4.1
Confidence intervals are a way to make predictions about entire populations using a random
sample. Suppose you do not know the value for a certain population mean, such as the average
miles per gallon for some new brand of car. You don’t have the data for every single car of this
new brand, so you can’t calculate the population mean. However, you can try to estimate it.
Instructions
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 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
4.2
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?
For this assignment answer questions and solve problems involving the null and alternative
hypotheses.
Instructions
1. Answer the following questions in a Word document:
• 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.
4.3
In this assignment, you will answer questions and solve problems involving the last three steps of
a hypothesis test: calculating a test statistic (step three), calculating the p-value (step four), and
finally, making the correct conclusion (step five).
Instructions
1. 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 hypotheses 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?
4.4
1. Answer true or false for each part, and if false, explain your answer.
a. The point estimate for the population mean, µ, of an x distribution is x-bar,
computed from a random sample of the x distribution.
b. Every random sample of the same size from a given population will produce
exactly the same confidence interval for µ.
c. For the same random sample, when the confidence level is reduced, the
confidence interval for µ becomes wider.
2. Use tables or software to find the t-value for a 90% confidence interval when the sample
size is 20.
3. A random sample of size 64 has sample mean 24 and sample standard deviation 4.
d. Is it appropriate to use the t distribution to compute a confidence interval for the
population mean? Why or why not?
e. Construct a 95% confidence interval for the population mean.
f. Explain the meaning of the confidence interval you just constructed.
4. How much do adult male grizzly bears weigh in the wild? Six adult males were captured,
tagged and released in California and here are their weights:
480, 580, 470, 510, 390, 550
g. What is the point estimate for the population mean?
h. Construct at 90% confidence interval for the population average weight of all
adult male grizzly bears in the wild.
i. Interpret the confidence interval in the context of this problem.
5. After going to a fast food restaurant, customers are asked to take a survey. Out of a
random sample of 340 customers, 290 said their experience was “satisfactory.” Let p
represent the proportion of all customers who would say their experience was
“satisfactory.”
j. What is the point estimate for p?
k. Construct a 99% confidence interval for p.
l. Give a brief interpretation of this interval.
6. In your own words, define each of the following terms that are used in hypothesis testing:
a.
b.
c.
d.
e.
f.
The null hypothesis.
The alternative hypothesis.
The test statistic
The p-value
Type I Error
Type II Error
7. Suppose the p-value for a right-tailed test is .0245.
a. What would be your conclusion at the .05 level of significance?
b. What would the p-value have been if it were a two-tailed test?
8. A random sample has 42 values. The sample mean is 9.5 and the sample standard
deviation is 1.5. Use a level of significance of 0.02 to conduct a left-tailed test of the
claim that the population mean is 10.0.
a.
b.
c.
d.
Are the requirements met to run a test like this?
What are the hypotheses for this test?
Compute the test statistic and the p-value for this test.
What is your conclusion at the 0.02 level of significance?
9. MTV states that 75% of all college students have seen at least one episode of their TV
show “Jersey Shore”. Last month, a random sample of 120 college students was selected
and asked if they had seen at least one episode of the show. Out of the 120, 85 of them
said they had seen at least one episode. Is there enough evidence to claim the population
proportion of all college student that have watched at least one episode is less than 75%
at the 0.05 level of significance?
a.
b.
c.
d.
Are the requirements met to run a test like this?
What are the hypotheses for this test?
Compute the test statistic and the p-value for this test.
What is your conclusion at the 0.05 level of significance?
...