Math_200_Project2_Jihee Yu
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Ivy Tech Community College, Northcentral *
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Course
200
Subject
Mathematics
Date
Apr 3, 2024
Type
docx
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Uploaded by JudgeWaspPerson825
2023_Math200 : jihee yu
Title of Project 2: "Analysis of Adult Female Heights and Comparison to Athletic Populations"
Introduction
The study of human height has been a subject of interest and research for
decades. It is a physical characteristic that varies widely within
populations, and understanding the distribution of heights can provide
valuable insights into various aspects of human biology and society. In
this project, we will conduct an analysis of adult female heights using a
dataset containing a random sample of 30 individuals.
The dataset consists of 30 adult female names and their corresponding
heights in inches. We will explore and analyze this data to gain a better
understanding of the distribution of heights among the sampled
individuals. Additionally, we will utilize statistical techniques and concepts
to
draw
meaningful
conclusions
from
this
dataset.
Normal Distribution & Sampling Distribution
1.
Tallest person z-score A.
Find the tallest person
: According from the data we can find the the tallest girl is Sophia
. Her height is 73.89.
z-score formula
is: z
=
x
−
μ
σ
. x is 73.89 and μ is the mean and σ is
standard deviation. But we are not going to use the exact value to
find the z score. So the mean would be 65 and the standard
deviation would be 3.5. z
=
73.89
−
65
3.5
=2.54.
This means Sophia’s height is 2.54 standard deviations over the given average female height.
B.
probabilities and interpretations
The probability that a randomly selected female is taller than she:
P(X≥73.89)= 1−P(X≤73.89)=1−P(Z≤2.54)=0.0055
The probability that a randomly selected female is shorter than she:
P(X≤73.89) = P(X≤73.89) = P(≤2.54) = 0.9945
C.
Is her height “unusual”?
Yes, Her eight is unusual. It is because the probability in B is 0.0055 < 0.05.
2. Shortest person z-score
A. Find the tallest person Professor :
Jerry J. Tauber
2023_Math200 : jihee yu
Title of Project 2: "Analysis of Adult Female Heights and Comparison to Athletic Populations"
We know z score formul is: z
=
x
−
μ
σ
. The shortest girl is Lillian in the data. Her height is 59.31. Then x
is 59.31. μ is the mean and σ is standard deviation.
z
=
59.31
−
65
3.5
= -1.63
Her z-score is -1.63, indicating that her height is 1.63 standard deviations below the mean.
B.
probabilities and interpretations
The probability that a randomly selected female is taller than she:
P(X≥59.31)= 0.9484
According to the data, probability is = 0.9484*100= 94.84% probability which is a randomly selected who is taller than her.
The probability that a randomly selected female is shorter than she:
P ( X ≤ 59.31 ) = 0.052
According to the data, probability is =0.052*100=5.2% This means there have 5.2% of chance that a randomly selected female is shorter than her. C.
Is her height “unusual”?
Based on this criterion, Lillian's height of 59.31 inches is not considered unusual. It falls within a reasonably expected range given the mean and standard deviation of the data. And also if it wants to satisfy “unusual” statement. It’s probability has to same or less than 0.05. 3. Find the shortest person from the data and using the population mean and standard deviation
We can use central limit theorem. ˙
X N
(
μ,
σ
√
n
)
=
0.63901
To understand the sampling distribution of the mean with a given
sample size, we can apply the Central Limit Theorem. It tells us that the
sample mean (X) follows a normal distribution with a mean (μ) and
standard deviation (σ) divided by the square root of the sample size (n). In
this specific case, the sample mean is approximately 0.63901, and it is
distributed normally.
4. Boxplot and report of outliers
Professor :
Jerry J. Tauber
2023_Math200 : jihee yu
Title of Project 2: "Analysis of Adult Female Heights and Comparison to Athletic Populations"
In the project dataset provided,
we are interested in
understanding the distribution
of heights among adult
females. As part of the
analysis, a boxplot was
constructed to visualize the
heights of the 30 individuals
sampled. This boxplot, when
interpreted in conjunction with
the project's objective, is
instrumental in identifying
outliers within the dataset. Outliers are data points that
significantly differ from the rest
of the dataset. They can be
either exceptionally high or
exceptionally low values when
compared to the central
distribution. In the context of
the project data, two outliers were identified in the boxplot: heights of 72.44 inches and 73.89 inches. These heights exceeded the typical range found within the interquartile range (IQR) and thus qualify as outliers.
5. confidence interval for the average heights of woman.
The resulting interval
reveals that the upper limit
is 66.51 inches, while the lower
limit is 64.29 inches.
Interpretation: With 95%
confidence, we can assert
that the average height of women falls within the range of 64.29 to 66.51 inches. This information supports the given mean of 65 inches because it falls within the computed upper and lower limits of the confidence interval.
6. Testing Whether Female Volleyball Players Are Unusually Tall.
Indeed, the average female volleyball player can be categorized as unusually tall. The upper limit for the average height of women, established with a 95% confidence interval, is 66.51 inches, while Professor :
Jerry J. Tauber
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2023_Math200 : jihee yu
Title of Project 2: "Analysis of Adult Female Heights and Comparison to Athletic Populations"
the lower limit is 64.29 inches. Comparatively, the average height of
a female volleyball player is 67 inches, which exceeds the upper limit by 0.49 inches. This clear deviation beyond the upper limit, supported by the 95% confidence level, indicates that the average female volleyball player is statistically considered unusually tall.
7. Hypothesis Test: Are Women Significantly Taller Than Average Soccer Players?
Null Hypothesis (H0): The average female height is equal to 63.7 inches.
Alternative Hypothesis (HA): The average female height is not equal to 63.7
inches.
Sample Mean (X): 65.4
inches Test Statistic
(Z-stat): 2.991 P-value: 0.0028
Given that the p-value is less than the significance level of 0.05, we reject the null hypothesis. Consequently, we can confidently conclude that there is sufficient evidence to suggest that the average female height is significantly greater than the average female soccer player's height."
Professor :
Jerry J. Tauber