STAT*2060DE F Questions for Assignment #5 | Complete Solution

STAT*2060DE F
Questions for Assignment #5

Assignment #5 will be another D2L quiz. Please input your responses before the deadline. Here are the questions. The usual rules apply (3 decimal places, etc.). Suppose we wish to test whether the mean of a certain population is equal to 80, against a two-sided alternative hypothesis. This population is normally distributed, and it is known that = 15:0. A simple random sample of 18 values revealed a sample mean of 84.8.

#1. What is the value of the appropriate test statistic?
#2. What is the p-value of the hypothesis test?
#3. Is there signi cant evidence against the null hypothesis that the population mean is
80, at the 5% level of signi cance?
A) Yes B) No
#4. Suppose in reality the population mean is actually 100 for this population. What type
of error did we make in the previous question?
A) Type I error.
B) Type II error.
C) Both errors.
D) Neither error.
The next several questions refer to the following information.
A large orthodontist practice is investigating a new supplier for a dental bonding agent.
One important consideration is the breaking strength of the agent. It is well known from a
very large amount of past experience that the agent currently in use has a mean breaking
strength of 6 Mpa. The new supplier is o ering a new type of agent, one that would result
in very large savings. One factor the orthodontist wants to investigate is the breaking
strength of the new agent. He uses the agent to ax a dental appliance on each of 12
extracted teeth, and then measures the breaking strength. He nds that the breaking
strength has a sample mean of 4.8 Mpa with a sample standard deviation of 1.1 Mpa.
For such a small sample size (n = 12), it is a bit dubious to use the t procedures, as
we cannot rely on the Central Limit Theorem. But let's assume a normally distributed
population here. Let's also assume that these observations can be thought of as a random
sample from the population of interest. That's also a bit dubious, but let's go with it.
Test the null hypothesis that the population mean breaking strength of the new agent is
equal to 6.0 Mpa, against a two-sided alternative hypothesis.
#5. What is the value of the appropriate test statistic?
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#6. What is the p-value of the appropriate test?
A) p 􀀀 value > :10.
B) :05 < p 􀀀 value < :10.
C) :025 < p 􀀀 value < :05.
D) :01 < p 􀀀 value < :025.
E) :002 < p 􀀀 value < :01.
#7. Is there strong evidence that the population mean breaking strength of the new agent
is less than 6.0 Mpa?
A) Yes B) No
A recent study by the Center for Science in the Public Interest found that movie theatre
popcorn can have an extremely high calorie count, as well as very large amounts of saturated
fat. The saturated fat content varied by a large amount from chain to chain. A large
popcorn at Regal theatres in the U.S. was found to contain 60 grams of saturated fat. By
contrast a Baconator hamburger from Wendy's (two beef patties + 6 slices of bacon) has a
puny 26 grams of saturated fat . Cineplex Odeon pops their popcorn in canola oil, resulting
in a much lower saturated fat level. However, the calorie content is still very high.
On top of the poor stated nutritional characteristics, the group found that the containers
actually contained much higher calorie and fat content than was claimed. Suppose you
wish to investigate the calorie content of large bags of popcorn at your local theatre.
You nd that the movie theatre chain claims that a large untopped bag of their popcorn
contains 920 calories on average. By going at di erent randomly selected times, you obtain
a sample of 24 bags. Suppose that it's reasonable to think of these 24 bags as a simple
random sample. You have these bags analyzed and that they contained 1210 calories on
average, with a sample standard deviation of 75 calories. A normal quantile-quantile plot
and boxplot illustrate the data:
For perspective, the theatre's claimed value of 920 calories is represented by a line in the
boxplot.
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#8. Construct a 95% con dence interval for the population mean calorie content of bags
of this type. Which one of the following is the appropriate interval?
A) 1210 30:00
B) 1210 31:67
C) 1210 32:82
D) 1210 32:99
E) 1210 34:43
You decide to carry out a hypothesis test. Since before the sample was drawn you thought
that, if anything, the company was understating the calorie content, you decide to use a
one-sided alternative hypothesis. You wish to test the null hypothesis that the population
mean calorie content is what the company claims it to be. You feel that for the given
scenario, a choice of signi cance level of = :05 is appropriate.
#9. Which one of the following best represents the hypotheses of the test?
A) H0 : = 920;Ha : > 920
B) H0 : = 920;Ha : 6= 920
C) H0 : 6= 920;Ha : = 920
D) H0 : = 1210;Ha : > 1210
E) H0 : = 1210;Ha : 6= 1210
#10. Would it be more appropriate to use a t procedure or a z procedure in this situation?
A) z B) t
#11. What is the value of the appropriate test statistic?
#12. The p-value of the appropriate test is closest to which one of the following?
A) 0
B) .02
C) .05
D) .10
E) 1
#13. Which one of the following statements is true?
A) The evidence against the null hypothesis is signi cant at the 1% level.
B) The evidence against the null hypothesis is signi cant at the 5% level, but not at the
1% level.
C) The evidence against the null hypothesis is signi cant at the 10% level, but not at the
5% level.
D) The evidence against the null hypothesis is not signi cant at the 10% level.
E) None of the above.
#14. Which one of the following statements is the best conclusion of the hypothesis test?
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A) There is signi cant evidence that the company's claim is true.
B) There is signi cant evidence that the population mean is greater than the company's
claim.
C) There is signi cant evidence that the sample mean is greater than the company's claim.
D) There is signi cant evidence that the sample mean is greater than the population mean.
E) There is not signi cant evidence against the company's claim.
The next series of questions refer to the following information.
Suppose that we are interested in buying ball bearings from a certain manufacturer. We
require very precisely manufactured ball bearings, and so we decide to investigate this
manufacturer's output. The manufacturer insists that at least 98% of their manufactured
bearings will be within our acceptable tolerance limits. We decide to test this claim.
Let's test the claim that the population proportion that are within tolerance limits is .98,
against the alternative that the true proportion is less than .98.
Suppose we draw a random sample of 1000 of the ball bearings, and nd only 912 are
within the acceptable tolerance limits.
#15. First, let's calculate a con dence interval for the population proportion. What is a
95% con dence interval for the proportion of bearings that are within tolerance limits?
A) :912 :018
B) :912 :021
C) :912 :024
D) :912 :027
E) :912 :030
Now on to the hypothesis test.
#16. Which one of the following represents the appropriate hypotheses?
A) H0 : p = :98;Ha : p > :98
B) H0 : ^p = :98;Ha : ^p < :98
C) H0 : p = :98;Ha : p 6= :98
D) H0 : p = :98;Ha : p < :98
#17. What is the value of the appropriate test statistic?
#18. What is the p-value of the test? (Choose the closest value)
A) .0001
B) .01
C) .05
D) .10
E) .15
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#19. Which one of the following statements is true?
A) The evidence against the null hypothesis is signi cant at the 1% level.
B) The evidence against the null hypothesis is signi cant at the 5% level, but not at the
1% level.
C) The evidence against the null hypothesis is signi cant at the 10% level, but not at the
5% level.
D) The evidence against the null hypothesis is not signi cant at the 10% level.
E) None of the above.
#20. Which one of the following is the best conclusion at the .05 signi cance level?
A) There is signi cant evidence that the population proportion within tolerance limits is
less than .98.
B) There is signi cant evidence that the sample proportion within tolerance limits is less
than .98.
C) There is not signi cant evidence that the population proportion within tolerance limits
is greater than .98.
D) There is signi cant evidence that the population proportion within tolerance limits is
greater than .98.
#21. Which of the following statements are true? (Check all that are true)
(Assume two-sided alternative hypotheses in all cases)
A) If we reject the null hypothesis at the 5% signi cance level, it must also be rejected at
the 10% signi cance level.
B) If we reject the null hypothesis at the 5% signi cance level, it would also be rejected at
the 1% signi cance level.
C) If we do not reject the null hypothesis at the 10% signi cance level, we will de nitely
not reject it at the 5% signi cance level.
D) If the p-value is large (say .9992) there is no evidence against the null hypothesis.
E) The p-value will be equal to 1 whenever the sample mean equals the hypothesized mean.
Consider the following output, summarizing the results of the t procedures for a one-sample
data set. One Sample t-test
One Sample t-test
data: rnorm(200, 50, 1)
t = 0.2936, df = 199, p-value = 0.7694
alternative hypothesis: true mean is not equal to 50
95 percent confidence interval:
49.88753 50.15182
sample estimates:
mean of x
50.01967
You should be able to see that the alternative hypothesis is Ha : 6= 50: The p-value for
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this alternative is given in the output.
#22. What would the p-value have been if the alternative were Ha : < 50.
#23. What would the p-value have been if the alternative were Ha : > 50.
(Hint: The answers to #22 and #23 are not equal.)
#24. How many of the following are based on sample data?
The signi cance level of a test.
The hypothesized mean.
The choice of alternative hypothesis.
A) 0
B) 1
C) 2
D) 3
#25. Which of the following statements are true? There may be more than one true
statement.
(Assume two-sided alternative hypotheses in all cases)
A) A test statistic can be negative.
B) A p-value can be negative.
C) A hypothesized mean can be negative.
D) A signi cance level can be negative.
E) If the Z statistic is equal to 1, then the p-value will also equal 1.