STAT*2060DE Assignment #4 | Complete Solution

STAT*2060DE

Questions for Assignment #4

Please input your responses into the D2L quiz \Assignment #4" before the deadline. As
always, give your responses to at least 3 decimal places where applicable.
The rst 4 questions refer to the following information.
In some locations, in order to label a product trans-fat free, the product must contain
less than .50 grams of trans-fat per serving. Suppose that Freedom's frozen french fries
are labelled as trans-fat free. A random sample of 120 servings of this type of french fry
yielded a sample mean trans fat level of .817 grams per serving. Suppose it is reasonable
to assume that the population standard deviation is .14 grams.
#1. What is the 90% con dence interval for the population mean trans fat level per serving
for Freedom's fries?
(For any con dence interval questions, on the quiz I may ask you for the interval itself, or
perhps just the margin of error. Please keep track of both of these values.)
A) (.790, .844)
B) (.793, .841)
C) (.796, .838)
D) (.799, .835)
E) (.802, .832)
#2. What is the 95% con dence interval for the population mean trans fat level per serving
for Freedom's fries?
A) :817 :022
B) :817 :025
C) :817 :028
D) :817 :031
E) :817 :034
#3. Does this interval provide us with strong evidence that the population mean trans-fat
per serving for Freedom's fries is greater than .50?
A) Yes. B) No.
#4. Suppose we want to estimate the population mean trans fat per serving to within .004,
with 95% con dence. How large a sample size would be required?
#5. Suppose we have a large sample of n = 400, we happen to know , and we use
x1:43=
p
n to obtain a con dence interval for . What would the con dence level of the
interval be? Express your response as a percentage, but please do not include the % sign.
For example, if the correct con dence level is 93.42%, input 93.42 into the quiz.
1
#6. What is the appropriate z =2 value for a 45.8% con dence interval?
#7. Suppose we wish to use the t procedure to construct a con dence interval for the
population mean. Under which of the following conditions would the methods perform
worst?
A) A small sample size and a skewed distribution.
B) A large sample size and a skewed distribution.
C) A small sample size and a symmetric distribution.
D) A large sample size and a symmetric distribution.
E) The t procedure would perform poorly in all of the above situations.
#8. Which of the following statements are TRUE? Check all that are true.
A) As the con dence level decreases (from 95% to 90%, say), the width of the interval
increases.
B) As the con dence level increases, the margin of error increases.
C) As the con dence level increases, the standard error of the sample mean decreases.
D) The margin of error decreases as the standard deviation decreases.
#9. Which of the following statements are TRUE? Check all that are true.
A) The t distribution gets closer to the standard normal distribution as the degrees of
freedom increase.
B) We use the t distribution in con dence intervals for the population mean when the
population variance is known.
C) The standard normal distribution has less area in the tails than the t distribution.
D) Very large values (bigger than 4, say) are more likely under the z distribution than the
t distribution.
E) The mean of the t distribution is greater that of the standard normal.
F) The variance of the t distribution is greater that of the standard normal. (N.B. I haven't
given you a formula for the variance of the t distribution, but you should be able to gure
this one out using basic logic and your knowledge of the relationship between the t and z
distributions).
#10. Suppose we should be using the t distribution to calculate a 90% con dence interval,
but instead we use the z distribution. Will the interval based on the z distribution be wider
or narrower than the correct interval using t?
A) Wider B) Narrower
#11. It's known that health care workers who use latex gloves with glove powder on a
daily basis are particularly susceptible to developing a latex allergy. Each in a sample of
46 hospital employees who were diagnosed with latex allergy based on a skin-prick test
reported on their exposure to the latex gloves (Current Allergy & Clinical Immunology,
March 2004). Summary statistics for the number of latex gloves used per week are the
sample mean 19.3 and sample standard deviation 11.9.
2
Assume for the purposes of this question the 95% con dence interval found is (11.1, 27.5).
Which one of the following is the most appropriate interpretation of this interval?
A) 95% of health care workers with a latex allergy wear leather gloves.
B) We can be 95% con dent that population mean number of latex gloves used per week
by health care workers with a latex allergy lies between 11.1 and 27.5.
C) 95% of health care workers with a latex allergy use between 11.1 and 27.5 gloves per
week.
D) We can be 95% con dent that the sample mean of the 46 employees lies between 11.1
and 27.5.
#12. Suppose we wish to estimate the population mean delay time for
ights leaving from
a large airport under reasonable weather conditions. What is the minimum sample size
that would be required if we wished to estimate the true mean delay time at this airport
to within 1.5 minutes with 90% con dence? Assume that = 13:0.
N.B. The next several 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. In a random sample of 400 of the ball bearings, 48 of them contain
manufacturing defects that make them useless to us.
#13. Is it reasonable to use the normal approximation to construct a con dence interval
for the population proportion in this situation?
A) Yes B) No
#14. What is the best estimate of the standard deviation of the sampling distribution of
the sample proportion in this scenario?
#15. What is a 95% con dence interval for the population proportion of defective balls?
A) :12 0:0288
B) :12 0:0298
C) :12 0:0308
D) :12 0:0318
E) :12 0:0328
#16. What is the 99% margin of error? (The margin of error is the \stu " to the right of
the +=􀀀 sign in the con dence interval formula, z =2
q
^p(1􀀀^p)
n in this case)
#17. What is the minimum sample size that would be required if wished to estimate the
true proportion of ball bearings with defects to within .001 with 90% con dence?
(Use the estimate ^p = 48=400 = :12 as the best estimate of p in the minimum sample size
formula)
#18. Suppose that we obtain a random sample of 1 ball bearings from this manufacturer,
3
and nd a 95% con dence interval for p, the proportion of defective bearings, to be (.110,
.130). Which of the following statements are true?
(There may be more than one true statement; check all that are true)
A) If we repeatedly sampled from the population of ball bearings, 95% of the time the
sample proportion would lie between .110 and .130.
B) If we repeatedly sampled from the population of ball bearings and calculated a 95%
con dence interval for p for each sample, 95% of these intervals would contain the true
proportion of defective ball bearings.
C) We can be 95% con dent that the sample proportion of defective bearings lies between
.110 and . 130.
D) 95% of the ball bearings chosen in the sample would be defective between 11% and 13%
of the time.
E) We can be 95% con dent we do not have a biased sample.
The next 4 questions refer to the following information.
Consider the following sample of size n = 9 from a normally distributed population:
3.8, 8.4, 8.2, 17.2, 11.9, 12.4, 5.7, 11.1, 12.3
#19. What is the point estimate of the population mean ?
#20. What is the standard error of the sample mean?
#21. What is the best estimate of the standard deviation of the sampling distribution of
the sample mean?
#22. What is a 95% con dence interval for the population mean ?
A) 10:111 2:637
B) 10:111 2:729
C) 10:111 2:833
D) 10:111 2:961
E) 10:111 3:103
4