BUQU 1230 Assignment on Confidence Interval

Extra Assignment # 1 / Spring, 2020
BUQU 1230 – Extra Assignment #1 (In Partial Replacement of Midterm & Final Exam)
Instructions:
1. Type your answers into this document. Save your work and upload it back to the course
Moodle website. Please keep in mind I won’t see your Excel work, so document all the
work you want me to see in this file.
2. You are both encouraged and permitted to use Excel to the utmost of your ability for this
assignment. This is an open textbook / open Internet assignment. You may find the
formula sheet on our Moodle website especially useful.
3. This assignment is due on Tuesday, March 31, 2020.
Question 1 ___ out of 11
Question 2 ___ out of 5
Question 3 ___ out of 10
Total ___ out of 26

Extra Assignment # 1 / Spring, 2020
Solutions
1. (11 points)
The province of British Columbia has recently grown concerned about the extent to
which its bicycle helmet law is being observed (or flouted.) They sent a few observers to
stand on bike routes throughout Vancouver and count bicyclists and note whether or not
they’re wearing a helmet. They found that of 200 cyclists observed, 168 were wearing
bike helmets. They passed the information along to you and asked you to assess the
situation in the population.
a. Use the sample data (given above) to generate a point estimate for the population
parameter of the proportion of Vancouver cyclists who wear a helmet. (1 point)
Solution
The population sample size = 200
The number of successes = 168
The point estimate of the population proportion is given by:
^p= x
n =168
200 =0.84
^p=0.84
b. Construct a 95% confidence interval around the population parameter you
calculated in part a). (3 points)
Solution
The critical value for ∝=0.05 is Zc=Z1−∝/ 2=1.96
The confidence interval is given by:
CI =( ^p−Zc √ ^p ( 1− ^p )
n , ^p+ Zc √ ^p ( 1− ^p )
n )

Extra Assignment # 1 / Spring, 2020
CI =( 0.84−1.96 √ 0.84 ( 1−0.84 )
200 , 0.84 +1.96 √ 0.84 ( 1−0.84 )
200 )
CI =( 0.789, 0.891)
The lower limit of the confidence interval is 0.789 while the upper limit of the
confidence interval is 0.891. It can alternatively be expressed as:
CI =0.789 ≤ ^p ≤ 0.891
c. Interpret the confidence interval constructed in part b). (2 points)
Solution
The confidence interval above means that at 95% level of confidence, the
actual point estimate of the proportion of Vancouver cyclists who wear a
helmet is between 0.789 and 0.891
d. Let’s say the province wants a confidence interval that’s no bigger than 5% (e.g. a
margin of error that’s no bigger than 2.5%). How many people would you
suggest they sample? (3 points)
Solution
The margin of error required is 0.025 while significance level is 0.05.
The point estimate for proportion selected for this scenario is 0.05 indicating
the worst case. The critical value for ∝=0.05 is Zc=Z1−∝/ 2=1.96 .The
minimum sample size is given by:
n ≥ p ( 1−p ) ( Zc
E )
2
n ≥ 0.5 ( 1−0.5 ) ( 1.96
0.025 )
2
=1536.58

Extra Assignment # 1 / Spring, 2020
The minimum sample size needs to satisfy the condition n ≥ 1536.58 and must
be an integer. Therefore, I would suggest they sample a minimum sample size
of 1537 cyclists.
e. The provincial bureaucrats look at your answer to part d and says, ‘that’s too
many people.’ They go on to ask, ‘is there any way we could arrive at a smaller
confidence interval WITHOUT taking a larger sample?’ What do you tell them?
(1 point)
Solution
Yes, there would be other way of they could arrive at a smaller confidence
interval without taking a larger sample. Some of the methods they could
apply include:
 Use a lower confidence level which would give a narrower and more
precise confidence interval.
 Reducing the variability of the data since the lesser the variability the
precise it is to estimate a population parameter (Selvanathan &
Keller, 2017).
 Use one sided confidence interval since it has a smaller margin of
error.
f. Given the way the sample was taken, do you see any problems with using the
sample data to estimate the situation in the overall population of cyclists in B.C.?
If so, what are they? (1 point)