Section 7 Lab - Megan McCauley
pdf
keyboard_arrow_up
School
Clemson University *
*We aren’t endorsed by this school
Course
3090
Subject
Statistics
Date
Apr 3, 2024
Type
Pages
5
Uploaded by PresidentPencilFrog40
STAT
3090
S
ECTION 7
L
AB S
PRING 2024 S
AMPLING D
ISTRIBUTIONS
1 N
AME
:
Megan McCauley
O
BJECTIVES
:
Upon successful completion of this assignment, you will be able to…
•
Recognize and use different sampling methods •
Describe the distribution of the sample mean and the sample proportion •
Understand the Central Limit Theorem and its application •
Understand the impact of Sample size on Sampling Variability •
Interpret the probability of sampling distribution Part One: Sampling Techniques 1.
A Beary Interesting Problem
(18 points)
To determine the effectiveness of jazz music in terms of bear attack prevention, a few students want to set up a study where they casually approach 7 bears while playing jazz music and then approach 7 bears with no music playing. Thus, they will need a sample of 14 bears. The students are near a wilderness area with 6 regions and they are also near to a zoo that has 5 bears. The 6 wilderness regions are approximately the same size and the number of bears in each area is approximately the same. Explain how the students could use each of the different types of sampling methods to gather their data. Note that some of the sampling methods would be nearly impossible to do. That is ok. Simply come up with a way they could use that particular method even if it is completely impractical. (Have fun with this) Disclaimer: Please do not try this experiment. a.
Simple random sampling (3 pts) Assign a number to every bear acrpss all 6 regions and the zoo, then put all the numbers on a spinning wheel and spin the wheel 14 times to get 14 bears. b.
Stratified random sampling (3 pts) The two strata are the bears from the wilderness regions and the bears from the zoo. Randomly select 1 bear from each wilderness region, then select 3 additional bears across all the wilderness regions, and include all 5 zoo bears in the study. c.
Cluster sampling (3 pts) The clusters are each individual wilderness region and the zoo; there are 7 clusters total. Randomly select two bears from each cluster. d.
Systematic sampling (3 pts) Let’s say there are 23 total bears in the wilderness regions. 23+5 zoo bears is 28. 28/14=2, which is the sampling interval. Select every 2
nd
bear until 14 bears have been picked. e.
Convenience sampling (3 pts) The students would use the 5 bears from the zoo and then 9 more bears in the region closest to them.
STAT
3090
S
ECTION 7
L
AB S
PRING 2024 S
AMPLING D
ISTRIBUTIONS
2 f.
Judgement sampling (3 pts) The students could go to the zoo and to each of the 6 regions and pick the 14 bears that they feel are most representative of the entire population by using metrics like age and color of fur. Sampling Distribution of Sample Mean 2.
Credit Card Debt
(10 points)
The mean credit card debt for a U.S. household is $7,115 with a standard deviation of $2,160. This mean is such a large value because of a few deeply indebted households. Consider a random sample of 50 U.S. households and let 𝑥̅
represent the sample mean credit card debt. Use this information to answer the following questions. (5 pts each) (a)
Describe the sampling distribution of sample mean , i.e., its shape (normal or not), mean, and standard error. (Keep 2 decimal places.) Mean: 2160 Standard deviation: 2160 Shape: normal Standard error = 𝜎
√𝑛
= 2160
√50
=305.18 N≥30 so normal
(b)
What is the probability that the mean credit card debt for a sample of 50 U.S. households is less than $6,500? Include probability notations and round your answer to 4 decimal places. Z= (6500−7115)
(
2160
√50
)
=
-2.017 𝑃(𝑧 < −2.017) =
0.0228 3.
Hot Dog Fat Content
(15 points)
A hot dog manufacturer claims that its new lean beef hot dogs have an average of 3.5g of fat per hot dog. The fat content of these hot dogs is known to be approximately normally distributed with a standard deviation of 0.5g. (5 pts each)
STAT
3090
S
ECTION 7
L
AB S
PRING 2024 S
AMPLING D
ISTRIBUTIONS
3 (a)
What is the probability that one of these lean beef hot dogs has more than 4g of fat? Include probability notations and round your answer to 4 decimal places.
𝑧 =
(𝑥−𝜇)
𝜎
=
(4−3.5)
0.5
= 1
𝑃(𝑧 > 1) =
0.1587 (b)
Describe the sampling distribution of the average fat content of a randomly selected package of 8 lean beef hot dogs, i.e., shape, mean, and standard error. Round to two decimal places if necessary. Shape: the sample size is less than 30, so the shape is abnormal Mean: 3.5g Standard error = 𝜎
√𝑛
=
0.5
√8
=
0.177 = 0.18 (c)
What is the probability that a randomly selected package of 8 lean beef hot dogs will have an average fat content greater than 4g? Use 4 decimal places in your final answer of the required probability and include probability notations. 𝑧 =
(𝑥−𝜇)
𝜎
=
(4−3.5)
0.177
=
2.825 𝑃(𝑧 > 2.825) =
0.0024 4.
WebAssign #1, 2, 3, 4, 8 (17 points)
****************************************************************************** Part 2: Sampling Distributions of sample proportion 5.
College Degrees
(9 points)
A survey is conducted from a population of people of whom 40% have a college degree. Of the 60 people who were surveyed, 27 indicated that they have a college degree. (a)
What is the sample proportion, 𝑝̅
, of respondents who have a college degree in this survey? Include the correct symbol for the sample proportion in your answer. (4pts) P=
27
60
= 0.45 (b)
Find the probability of obtaining a sample proportion less than or equal to the one that you calculated in part (a) from a sample of size 60. (5pts) To get full credit, you need to: •
Use the sampling distribution of sample proportion 𝑝̅
.
Your preview ends here
Eager to read complete document? Join bartleby learn and gain access to the full version
- Access to all documents
- Unlimited textbook solutions
- 24/7 expert homework help
STAT
3090
S
ECTION 7
L
AB S
PRING 2024 S
AMPLING D
ISTRIBUTIONS
4 •
Determine the shape, mean, and standard error of 𝑝̅
. •
Draw a sketch with appropriate area shaded. •
Use the proper probability notations. •
Get the correct answer with 4 decimal places. Shape: 60 people > 30, so the shape is normal Mean: population proportion = 0.40 Standard error: SE=
√(
0.40(1−0.040)
60
)
=
0.0632 Z = (0.45−0.40)
0.0632
=
0.791 P(Z
≤
0.791) = 0.7867 6.
Baking Shows (8 points)
A nationwide survey analyzing trends in popular media found that 81% of U.S. college students prefer British baking shows over American baking shows. You are interested to see if this result holds at your university. You take a sample of 140 students on campus and find that 125 of them prefer watching British baking shows. (a)
Will the sampling distribution of sample proportion, 𝑝̅
, of students at your university who prefer British baking shows follow a normal distribution? Check the appropriate conditions to explain your answer. (3 points)
Yes, it will follow a normal distribution because the sample was selected randomly, one student’s preference for British baking shows does not influence another students so the observations are independent, and n>30 so the sample size is large. (b)
What is the probability of obtaining a sample of size 140 where 125 or more students prefer British baking shows? Include proper probability notations and round your answer to 4 decimal places. (5 points) 𝜇 = 140 ⋅ 0.81
= 113.4 𝜎 = 4.685
Z=
(125−113.4)
4.685
=
2.477 𝑃(𝑍 ≥ 2.477) =
0.0060
STAT
3090
S
ECTION 7
L
AB S
PRING 2024 S
AMPLING D
ISTRIBUTIONS
5 7.
Park Visitors
(8 points) Leslie has been tasked with putting together a statistical report regarding the use of a particular park in her hometown. Previous data shows that 72% of the residents living in the town visited the park in the last month. She wants to know the probability that more than 98 individuals in a random sample of 150 residents have visited the park in the last month. (a)
Will the sampling distribution of sample proportion, 𝑝̅
, of residents who have visited the park in the last month follow a normal distribution? Check the appropriate conditions to justify your answer. (2 points) Yes, it’ll follow a normal a distribution because the sample is selected randomly, one resident’s visit to the park doesn’t influence another resident’s visit so it’s independent, and both np and n(1-p) are greater than 10. (b)
What is the lower bound you will use for 𝑝̅
for Leslie’s sample of 150 residents
to solve the problem? Round your answer to two decimal places. (2 points) SE = √
𝑝(1−𝑝)
𝑛
= √(
0.72(1−0.72)
150
)
=
0.0367 Lower bound = 0.72 - 1.96 * 0.0367 = 0.65 (c)
What is the probability that at least 98 individuals in a random sample of 150 residents have visited the park in the last month? Include proper probability notations and round your answer to 4 decimal places. Use the sampling distribution of 𝑝̅
to find the probability. (4 points) Z = (0.65−0.72)
0.0367
=
-1.907 𝑃(𝑧 ≥ −1.907) =
0.9717 8.
WebAssign #5, 6, 7, 9
(15 points)
Recommended textbooks for you
Holt Mcdougal Larson Pre-algebra: Student Edition...
Algebra
ISBN:9780547587776
Author:HOLT MCDOUGAL
Publisher:HOLT MCDOUGAL
College Algebra (MindTap Course List)
Algebra
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:Cengage Learning
Recommended textbooks for you
- Holt Mcdougal Larson Pre-algebra: Student Edition...AlgebraISBN:9780547587776Author:HOLT MCDOUGALPublisher:HOLT MCDOUGALCollege Algebra (MindTap Course List)AlgebraISBN:9781305652231Author:R. David Gustafson, Jeff HughesPublisher:Cengage Learning
Holt Mcdougal Larson Pre-algebra: Student Edition...
Algebra
ISBN:9780547587776
Author:HOLT MCDOUGAL
Publisher:HOLT MCDOUGAL
College Algebra (MindTap Course List)
Algebra
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:Cengage Learning