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SamplingDistribution of
a SampleMean
Robb T.Koether
HomeworkReview
Introduction
AnExperiment
Assignment
Sampling Distribution of a Sample MeanLecture 26Section 8.4
Robb T. Koether
Hampden-Sydney College
Mon, Oct 13, 2008
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SamplingDistribution of
a SampleMean
Robb T.Koether
HomeworkReview
Introduction
AnExperiment
Assignment
Outline
1 Homework Review
2 Introduction
3 An Experiment
4 Assignment
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SamplingDistribution of
a SampleMean
Robb T.Koether
HomeworkReview
Introduction
AnExperiment
Assignment
Homework Review
Exercise 8.12, page 528.
Suppose that 60% of all students at a large universityaccess course information using the Internet.(a) Sketch a picture of the distribution for the possible
sample proportions you could get based on a simplerandom sample of 100 students.
(b) Use the 68–95–99.7 rule for normal distributions tocomplete the following statements:
(i) There is a 68% chance that the sample proportion isbetween _____ and _____.
(ii) There is a 95% chance that the sample proportion isbetween _____ and _____.
(iii) It is almost certain that the sample proportion isbetween _____ and _____.
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SamplingDistribution of
a SampleMean
Robb T.Koether
HomeworkReview
Introduction
AnExperiment
Assignment
Homework Review
Exercise 8.12, page 528.
(c) Would it be likely to observe a sample proportion of0.50, based on a simple random sample of size 100, ifthe population proportion were 0.60? Explain.
(d) Sketch a picture of the distribution for the possiblesample proportions you could get based on a simplerandom sample of 400 students.
(i) How does this picture differ from the one in part (a)?(ii) How will the increased sample size affect the range of
values you gave in (i)–(iii) of part (b)
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SamplingDistribution of
a SampleMean
Robb T.Koether
HomeworkReview
Introduction
AnExperiment
Assignment
Homework Review
Solution(a) For n = 100 students, the sketch is
0.600.453 0.747
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SamplingDistribution of
a SampleMean
Robb T.Koether
HomeworkReview
Introduction
AnExperiment
Assignment
Homework Review
Solution(b) First, compute µp̂ and σp̂.
µp̂ = p = 0.60.
σp̂ =
√p(1− p)
n=
√(0.60)(0.40)
100= 0.0490.
(i) There is a 68% chance that the sample proportion isbetween 0.60− 0.0490 = 0.551 and0.60 + 0.0490 = 0.649.
(ii) There is a 95% chance that the sample proportion isbetween 0.60− 2(0.0490) = 0.502 and0.60 + 2(0.0490) = 0.698.
(iii) It is almost certain that the sample proportion is between0.60− 3(0.0490) = 0.453 and 0.60 + 3(0.0490) = 0.747.
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SamplingDistribution of
a SampleMean
Robb T.Koether
HomeworkReview
Introduction
AnExperiment
Assignment
Homework Review
Exercise 8.12, page 528.
(c) The question should ask how likely it is to observe asample proportion at least as low as 0.50. Theprobability is P(p̂ ≤ 0.50), which is
normalcdf(-E99,0.50,0.60,0.490) = 0.0206.
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SamplingDistribution of
a SampleMean
Robb T.Koether
HomeworkReview
Introduction
AnExperiment
Assignment
Homework Review
Exercise 8.12, page 528.
(d) For n = 400 students, the sketch of p̂ is
0.600.5265 0.6735
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SamplingDistribution of
a SampleMean
Robb T.Koether
HomeworkReview
Introduction
AnExperiment
Assignment
Homework Review
Exercise 8.12, page 528.
(d) (i) This distribution is only half as wide (and twice as tall).(ii) The standard deviation of p̂ is only half as much, so the
answers are(i) There is a 68% chance that the sample proportion is
between 0.60− 0.0245 = 0.5755 and0.60 + 0.0245 = 0.6245.
(ii) There is a 95% chance that the sample proportion isbetween 0.60− 2(0.0245) = 0.5510 and0.60 + 2(0.0245) = 0.6490.
(iii) It is almost certain that the sample proportion is between0.60− 3(0.0245) = 0.5265 and 0.60 + 3(0.0245) = 0.6735.
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SamplingDistribution of
a SampleMean
Robb T.Koether
HomeworkReview
Introduction
AnExperiment
Assignment
Introduction
Recall the experiment we did in which we collected 100samples of size 5 and computed the sampleproportions.We graphed our results and saw a good approximationto the normal curve.Then we calculated the mean and standard deviation ofour distribution and found that we had goodapproximations to
µp̂ = p
and
σp̂ =
√p(1− p)
n.
Now we will do the same with sample means.
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SamplingDistribution of
a SampleMean
Robb T.Koether
HomeworkReview
Introduction
AnExperiment
Assignment
Introduction
Recall the experiment we did in which we collected 100samples of size 5 and computed the sampleproportions.We graphed our results and saw a good approximationto the normal curve.Then we calculated the mean and standard deviation ofour distribution and found that we had goodapproximations to
µp̂ = p
and
σp̂ =
√p(1− p)
n.
Now we will do the same with sample means.
![Page 12: Sampling Distribution of a Sample Mean - Lecture 26 Section 8people.hsc.edu/faculty-staff/robbk/Math121/Lectures/Fall... · 2008-10-14 · Sampling Distribution of a Sample Mean Robb](https://reader033.vdocument.in/reader033/viewer/2022050411/5f87d8afdfe13f59534e8153/html5/thumbnails/12.jpg)
SamplingDistribution of
a SampleMean
Robb T.Koether
HomeworkReview
Introduction
AnExperiment
Assignment
Introduction
Recall the experiment we did in which we collected 100samples of size 5 and computed the sampleproportions.We graphed our results and saw a good approximationto the normal curve.Then we calculated the mean and standard deviation ofour distribution and found that we had goodapproximations to
µp̂ = p
and
σp̂ =
√p(1− p)
n.
Now we will do the same with sample means.
![Page 13: Sampling Distribution of a Sample Mean - Lecture 26 Section 8people.hsc.edu/faculty-staff/robbk/Math121/Lectures/Fall... · 2008-10-14 · Sampling Distribution of a Sample Mean Robb](https://reader033.vdocument.in/reader033/viewer/2022050411/5f87d8afdfe13f59534e8153/html5/thumbnails/13.jpg)
SamplingDistribution of
a SampleMean
Robb T.Koether
HomeworkReview
Introduction
AnExperiment
Assignment
Introduction
Recall the experiment we did in which we collected 100samples of size 5 and computed the sampleproportions.We graphed our results and saw a good approximationto the normal curve.Then we calculated the mean and standard deviation ofour distribution and found that we had goodapproximations to
µp̂ = p
and
σp̂ =
√p(1− p)
n.
Now we will do the same with sample means.
![Page 14: Sampling Distribution of a Sample Mean - Lecture 26 Section 8people.hsc.edu/faculty-staff/robbk/Math121/Lectures/Fall... · 2008-10-14 · Sampling Distribution of a Sample Mean Robb](https://reader033.vdocument.in/reader033/viewer/2022050411/5f87d8afdfe13f59534e8153/html5/thumbnails/14.jpg)
SamplingDistribution of
a SampleMean
Robb T.Koether
HomeworkReview
Introduction
AnExperiment
Assignment
The US Senate
There are 100 senators in the US Senate.Their tenures range from 1 year to 49 years.The mean and standard deviation of the population are
µ = 13.45 years
andσ = 11.18 years.
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SamplingDistribution of
a SampleMean
Robb T.Koether
HomeworkReview
Introduction
AnExperiment
Assignment
The US Senate
The histogram:
0
5
10
15
25
20
0 10 50403020
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SamplingDistribution of
a SampleMean
Robb T.Koether
HomeworkReview
Introduction
AnExperiment
Assignment
The US Senate
The boxplot:
0 10 50403020
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SamplingDistribution of
a SampleMean
Robb T.Koether
HomeworkReview
Introduction
AnExperiment
Assignment
An Experiment
State Years State Years State Years State Years State Years State Years State YearsAL 11 DE 7 IA 23 MI 7 NH 5 OK 14 TX 15AL 10 DE 35 IA 27 MI 29 NH 15 OK 3 TX 6AK 40 FL 7 KS 11 MN 1 NJ 2 OR 11 UT 31AK 6 FL 3 KS 12 MN 5 NJ 5 OR 12 UT 15AZ 13 GA 5 KY 23 MS 1 NM 25 PA 1 VT 1AZ 21 GA 3 KY 9 MS 30 NM 35 PA 27 VT 33AR 5 HI 18 LA 11 MO 1 NY 7 RI 1 VA 1AR 9 HI 45 LA 3 MO 21 NY 9 RI 11 VA 29CA 16 ID 17 ME 13 MT 1 NC 5 SC 5 WA 7CA 15 ID 9 ME 11 MT 30 NC 3 SC 3 WA 15CO 11 IL 11 MD 1 NE 7 ND 16 SD 11 WV 49CO 3 IL 3 MD 21 NE 11 ND 16 SD 3 WV 23CT 19 IN 31 MA 46 NV 7 OH 1 TN 1 WI 19CT 27 IN 9 MA 21 NV 21 OH 9 TN 5 WI 15
WY 1WY 11
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SamplingDistribution of
a SampleMean
Robb T.Koether
HomeworkReview
Introduction
AnExperiment
Assignment
An Experiment
State Years State Years State Years State Years State Years State Years State Years1 11 15 7 29 23 43 7 57 5 71 14 85 152 10 16 35 30 27 44 29 58 15 72 3 86 63 40 17 7 31 11 45 1 59 2 73 11 87 314 6 18 3 32 12 46 5 60 5 74 12 88 155 13 19 5 33 23 47 1 61 25 75 1 89 16 21 20 3 34 9 48 30 62 35 76 27 90 337 5 21 18 35 11 49 1 63 7 77 1 91 18 9 22 45 36 3 50 21 64 9 78 11 92 299 16 23 17 37 13 51 1 65 5 79 5 93 7
10 15 24 9 38 11 52 30 66 3 80 3 94 1511 11 25 11 39 1 53 7 67 16 81 11 95 4912 3 26 3 40 21 54 11 68 16 82 3 96 2313 19 27 31 41 46 55 7 69 1 83 1 97 1914 27 28 9 42 21 56 21 70 9 84 5 98 15
99 1100 11
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SamplingDistribution of
a SampleMean
Robb T.Koether
HomeworkReview
Introduction
AnExperiment
Assignment
An Experiment
Work in pairs.Use the TI-83 to get 10 samples of 5 senators each.(Allow repetitions.)For each sample, find the number of years that eachsenator has been in the senate.Record the average (out of 5).When you are finished, report the 10 sample meansthat you found.
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SamplingDistribution of
a SampleMean
Robb T.Koether
HomeworkReview
Introduction
AnExperiment
Assignment
An Example
For example,Sample Tenures Mean
{54, 38, 28, 70, 9} {11, 11, 9, 9, 16} 11.2{46, 84, 6, 72, 49} {5, 5, 21, 3, 1} 7.0{7, 4, 32, 26, 79} {5, 6, 12, 3, 5} 6.2{33, 18, 80, 56, 35} {23, 3, 3, 21, 11} 12.6{85, 54, 59, 25, 27} {15, 11, 2, 11, 31} 14.0{99, 73, 63, 82, 56} {1, 11, 7, 3, 21} 8.6{51, 20, 72, 46, 70} {1, 3, 3, 5, 9} 4.2{70, 1, 93, 87, 95} {9, 11, 7, 31, 49} 21.4{25, 5, 4, 28, 66} {11, 13, 6, 9, 3} 8.4{40, 73, 88, 1, 51} {21, 11, 15, 11, 1} 11.8
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SamplingDistribution of
a SampleMean
Robb T.Koether
HomeworkReview
Introduction
AnExperiment
Assignment
An Example
0
1
2
3
4
0 10 50403020
5
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SamplingDistribution of
a SampleMean
Robb T.Koether
HomeworkReview
Introduction
AnExperiment
Assignment
Assignment
HomeworkStudy the Central Limit Theorem.Catch up on past homework.