Download T_F_Ex2_F11

STP 226
FALL
2011
Instructor:
EXAM #2
Material from chapters 5-8
PRINTED NAME: ___________________________________________________
CLASS TIME:______________________________________
Honor Statement:
I have neither given nor received information regarding this exam, and I will not do so
until all exams have been graded and returned.
Signed ___________________________________
Date_________________________
DIRECTIONS:
This is a closed book examination. You may use an 8x11 page with hand written notes
(one side only) and a graphing calculator.
Formulas , z-tables and t-tables are included at the end of the test.
There are 8 problems. You can earn 104 points (4 points are extra credit points ).
For Question #1 there is no need to show work. For remaining 7 problems provide
complete and well-organized answers. Include sketches as requested to explain your
answers. Round up all answers to at least 4 decimal places.
If you use a calculator for any of the procedures (like computing confidence interval or
finding areas under normal or t curves), clearly indicate what calculator and what
procedure you used and what specific values you typed in.
Show your work!
RELAX and Good Luck!
Question #1 (2 points each )
Decide if each of the following statements is True or False.
a) Normal distribution curve with mean 16 and standard deviation 5 is wider than normal
curve with mean 16 and standard deviation 3.
True
False
b) If we compute 95 % and 90% confidence intervals for the mean final exam score of all
Mat 117 students at ASU last semester then 90% confidence interval will be wider than
95% confidence interval.
True
False
t
0.5 ,
c)For
for the t distribution curve with 7 degrees of freedom will be a
z
smaller number than
True
False
d) Margin of error in 90% confidence interval for
size.
True
decreases with increasing sample
False
e) 80th percentile of N(0,1) = z 0. 20
True
False
f) Consider the normal curve with mean 10 and standard deviation 4 . According to
Empirical Rule 68.26% of the area under that curve is between 6 and 14.
True
False
g)If normally distributed variable X has mean
=16 and standard deviation
=5 then
X− 5
has standard normal distribution.
16
True
False
h) Suppose the mean annual income for adult women in one city is $28,520 with standard
deviation of $5190 and the distribution is extremely left skewed. For the samples of size
99, x has approximately normal distribution.
True
False
Question #1 continues on the next page.
i) Suppose 90% confidence interval for a mean age of participants in a large mathematical
conference , based on a random sample of 120 participants, is (35, 49). We can say that
90% of people in the sample are between 35 and 49 years old.
True
False
j) If events A,B are mutually exclusive with P(A)=0.2 and P(B)=0.5, then P(A or B)=0.7
True
False
k) Using a simple random sample of size 5 we can compute a confidence interval for
mean age of some large population using z-interval procedure if we know population
standard deviation, but population is not normally distributed.
True
False
l) If eight decile of N(0,1) =A then second decile of N(0,1) = -A
True
False
m) If ̄ x is the sample mean for the sample of size n=16 from normal population with
x̄ − 20
mean 20 and standard deviation 8, then variable
has N(0,1) distribution
2
True
False
n) Sampling error in estimating population mean by the sample mean increases with
increasing sample size.
True
False
o) For normally distributed variable X with mean 10 and standard deviation 4 ,
P(X>16) =P(Z>1.5) , where Z has N(0,1) distribution.
True
False
p) Suppose sample of size=62 from some large population has ̄ x =12 and s=12, we can
use t-interval procedure for the confidence interval for the population mean even if
population has left skewed distribution.
True
False
q) Suppose probability of you passing this test is 0.75, then probability of you failing it is
25%.
True
False
Question #2 Medical Tests on Emergency Patients.
The frequency distribution shown below illustrates the number of medical tests
conducted on 36 randomly selected emergency patients.
Number of Tests
0
Number of patients 12
1
2
3
4 or more
8
5
5
6
Suppose one patient is selected at random, compute following probabilities, leave
answers in fraction form
(a)(4 points) Probability that this patient had exactly 2 tests done
ANSWER:___________________
(b) (4 pts) Probability that this patient had at least 1 tests and no more than 2 tests
done.
ANSWER:___________________
Question #3. College Degrees Awarded.
The relative frequency table below represents college degrees awarded in recent academic
year by gender (at ASU)
Bachelor's
Master's
Doctorate
0.310
0.101
0.013
Female 0.407
0.158
0.011
Male
Suppose a degree is randomly selected. Compute the following probabilities:
a) (4 points) Probability that the degree is not a master's degree.
ANSWER:___________________
b)(4 points) Probability that that degree is awarded to a man or it is a master's
degree
ANSWER:___________________
Question #4. Standard Normal Curve.
Use the tables of standard normal curve or a calculator to find the following.
Include appropriate sketch explaining each answer.
a. (4 points) area between - 1.32 and 1.19
ANSWER:___________________
b. (4 points) area left of
-2.15
ANSWER:___________________
c. (4 points) 58th percentile of the standard normal curve
ANSWER:___________________
Question #5. Chocolate Bar Calories.
The number of calories in a certain 1.5-ounce chocolate bar has normal distribution with
mean 225 with standard deviation of 10 calories.
Answer following questions, include appropriate sketch explaining your answer.
a) ( 4 points) What percentage of chocolate bars will have number calories exceeding
255 calories?
ANSWER:___________________
b)(4 points) How many calories are in a chocolate bar that is in 95th percentile of that
distribution?
ANSWER:____________________
Question #6 SAT Scores.
The national SAT test scores (for Verbal and Math)) have normal distribution with a
mean μ =1025 points and standard deviation σ = 90 points
a) (4 points)Let x be a sample mean for the samples of size 9 from all the SAT
scores. What is the sampling distribution of x ? Give mean and standard
deviation of that distribution.
Sampling distribution of
μ x = _______________
x
is __________________________________
σ x =_________________________
b)(4 points) Suppose we select a random sample of 9 SAT scares and
average. What is the probability that x be smaller than 940?
x
is their
P( x < 940)=__________________________________
Question #7 Body Temperature
A study reports that oral body temperature for a random sample of 30 healthy adults
living in AZ had a sample mean of 98.25 degrees Fahrenheit. Suppose we know that
population standard deviation is 0.73 degrees Fahrenheit.
a) (8 points)Find 80% Confidence Interval (CI) for μ =mean number of nights all AZ
residents stayed in a hotel during their vacations last year. Use z-interval procedure.
Give margin of error in your CI: ____________________
Compute both endpoints of your CI: _____________________
d) ( 4 points) What sample size is needed so that margin of error in our 80% CI will be
0.05?
n=_________________________________________
Question #8 Length of Children's Animated Films.
A data below represents lengths (in minutes) for a random sample of 12 popular
children's animated films. Let μ be the true average length of all children's animated
films. Assume normal distribution.
74, 76, 77, 78, 78, 76, 81, 83, 81, 83, 85, 92
a) (4 points) Compute a point estimate of μ
ANSWER:_________________________________
b)(4 points) Suppose you wanted to compute 90% confidence interval for μ and use a tinterval procedure. What is the appropriate t-value you need to use in computing of a
margin of error in your confidence interval, make sure to use appropriate degrees of
freedom.
Select correct answer:
A) t=1.363
B) t=1.796
C) t=1.372
D) t=1.812
ANSWER:_________________________________
c)(4 points) Suppose 95 % confidence interval for μ is: (77.2 min , 83.5 min ).
Based on that interval do you think it is reasonable to assume that μ is more than 1.25
hours (75 min)? Select appropriate answer from the following:
i) Yes, because both endpoints of the CI are above 75min.
ii) No, because 75 min is outside of the CI .
iii) Yes, because sample mean is more than 75 min
iv) No, because lower endpoint of CI is only few minutes above 75min
ANSWER:_________________________________
d) (4 points) Without computations will 99% CI for μ be wider then, narrower than or
the same as 90% CI for μ ? Select appropriate answer:
A) wider
B)
narrower
C) the same
FORMULAS
Sample statistics
x=
Sample mean:
∑ xi
, Sample standard deviation (definition) s =
n
∑ x 2i −
Computational formula
s=
∑ xi
∑
x i− x
n−1
2
n
n− 1
Population parameters:
=
Population mean:
∑ x 2i −
=
or
∑ xi
N
Population standard deviation:
=
∑
x i−
N
2
N
Standard score or z-score z =
x−μ
σ
If x ~ N
then z ~ N 0,1
,
Probability
P(E)=f/N P(A or B) = P(A) + P(B) - P(A and B)
Sampling Distribution of
x
=
,
x
=
x
x−
, Standardized version of x : z = / n
n
Confidence Intervals for
Confidence level C= 1−
Z-interval: x ± z
n=
z
/2
∗ 1 00 %
E= z
n Margin of error:
/2
n , Sample size estimation:
2
/2
E
t-interval: x ± t
/2
x−
s
x : t=
,
df=n-1,
Studentized
version
of
s/ n
n
2
2