statistics
vip500
rework_exam3_.pdf
STA 2303
Exam3 REWORK
Due nlt Tuesday 12/2
(at the beginning of class)
Name:
UTSA id:
Instructions: Show all of your work for full credit, and calculate probabilities to 4 decimal
places. Write your solutions below each problem. You may use additional paper, but keep the
solutions behind each respective problem. Use this page as your cover sheet. For full credit,
you must do all problems in their entirety. Points earned will be added to your Exam 3 grade.
This is an open book/note assignment. Good luck!
Consider an instrument that is used to measure temperature (°F) inside of an office building.
Such an instrument hangs on a wall just outside of Dilbert’s cubicle. Dilbert’s office building is
temperature-controlled, and set to a year-round mean temperature of 68°F, with a standard
deviation of 6°F.
Dilbert found an article online that claims that uncomfortable temperatures in the
workplace make unproductive workers. Specifically, it states that temperatures below 66°F or
above 71°F have been shown to make employees uncomfortable -- sleepy and/or grumpy.
Dilbert will be evaluated by his boss next month, and he now thinks that he will blame
his lack of productivity on uncomfortable temperatures in the office. He decides that the next
day he will meticulously check and record the temperature every 15 minutes for 8 hours. In this
sample of size 32, he finds that the average temperature is 66.4°F with a standard deviation of
6.2.
1) Identify the given parameters, statistics and sample size. Use correct notation!
2) What is the sampling distribution of the average temperature for all such samples of size 32?
Justify.
3) Compute the probability that the average temperature Dilbert will observe is considered
uncomfortable (thus causing his lack of productivity.)
4) Dilbert doubts that the year-round mean temperature in the office is 68°F, as is claimed by
the thermostat. He thinks that it must be lower, causing his excessive drowsiness. Test an
appropriate hypothesis at the .05 level of significance.
a. Write the null and alternative hypotheses.
b. State the null hypothesis in words.
c. What would be the implications of a Type I error in this case? What is the
probability of making this type of error?
d. Compute the statistic and define the critical region.
e. What is your conclusion, and interpretation?
5) Provide a 95% confidence interval for the true mean temperature. (Assume that population
parameters are unknown.) Interpret.
Determined to prove his point, Dilbert continues to collect data for a period of 30 workdays. He
records each day as “comfortable” if the average temperature for the day is between 66 and 71
degrees, and “uncomfortable” otherwise. The probability that a day will be uncomfortable has
already been computed in problem #3. In his sample of 30 days, only 2 days qualified as
uncomfortable.
6) Identify the population and sample proportions of uncomfortable days, using correct
notation.
7) Using sample data only, estimate the true proportion of uncomfortable days by computing a
95% confidence interval. Interpret. Does the CI include the parameter value?
8) During the evaluation, Dilbert tells his boss that his lack of productivity is due to
uncomfortable temperatures in the office, which affect at least 15% of his work days. Using
all the information above, do you think Dilbert makes a valid point? Explain your answer
using any of the above computations, and discuss your concerns about the data provided.
Z and t percentiles
Percentile Right Tail Area Z value t value 90% .10 1.28 1.31 95% .05 1.645 1.697
97.5% .025 1.96 2.042 99% .01 2.33 2.457
99.5% .005 2.575 2.75 Formulas 100x
(1 )% Confidence Intervals
for
p :
̂p Z / 2 ̂p (1 ̂p )/n
for
:
x t ( / 2);n1
s
n
Critical Regions for an
significance level hypothesis test
about
, known
H A
: o
Reject
H 0 if
x o Z
n
H A
: o
Reject
H 0 if
x o Z
n
H A
: o
Reject
H 0 if
x o Z / 2
n or
x o Z / 2
n
about
, unknown
H A
: o
Reject
H 0 if
x o t
( ) ;n 1
s
n
H A
: o
Reject
H 0 if
x o t
( ) ;n 1
s
n
H A
: o
Reject
H 0 if
x o t / 2
s
n OR
x o t / 2
s
n
