Statistics-I (395)
Q.1:a) Define the standard normal probability destiny function and the standard normal Cumulative distribution function. Give the equation of normal curve with mean 0 And
standard deviation 1.The single most important distribution in
probability and statistics is the normal probability distribution. The density
function of a normal probability distribution is bell shaped and symmetric
about the mean. The normal probability distribution was introduced by the
French mathematician Abraham de Moivre in 1733. He used it to approximate
probabilities associated with binomial random variables when n is large. This
was later extended by Laplace to the so-called CLT, which is one of the most
important results in probability. Carl Friedrich Gauss in 1809 used the normal
distribution to solve the important statistical problem of combining
observations. Because Gauss played such a prominent role in determining the
usefulness of the normal probability distribution, the normal probability
distribution is often called the Gaussian distribution. Gauss and Laplace
noticed that measurement errors tend to follow a bell-shaped curve, a normal
probability distribution. Today, the normal probability distribution arises repeatedly
in diverse areas of applications. For example, in biology, it has been observed
that the normal probability distribution fits data on the heights and weights
of human and animal populations, among others.
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We should also mention here that almost all
basic statistical inference is based on the normal probability distribution.
The question that often arises is, when do we know that our data follow the
normal distribution? To answer this question, we have specific statistical
procedures that we study in later chapters, but at this point we can obtain
some constructive indications of whether the data follow the normal
distribution by using descriptive statistics. That is, if the histogram of our
data can be capped with a bell-shaped curve (Fig. 3.2), if the stem-and-leaf
diagram is fairly symmetrical with respect to its center, and/or by invoking
the empirical rule “backward,” we can obtain a good indication of whether our
data follow the normal probability distribution.
The standard normal distribution (z
distribution) is a normal distribution with a mean of 0 and a standard
deviation of 1. Any point (x) from a normal distribution can be converted to
the standard normal distribution (z) with the formula z = (x-mean) / standard
deviation. z for any particular x value shows how many standard deviations x is
away from the mean for all x values. For example, if 1.4m is the height of a
school pupil where the mean for pupils of his age/sex/ethnicity is 1.2m with a
standard deviation of 0.4 then z = (1.4-1.2) / 0.4 = 0.5, i.e. the pupil is
half a standard deviation from the mean (value at centre of curve).
b) Find the value of X which corresponds
to a standardized value of -2.05 and 0.86 for each of the following
distributions.
i)
X N (62.3,38)
ii) X
N (m,s2 )
iii) X
N (a,b) (10+10)
Standard deviation in statistics, typically denoted by σ,
is a measure of variation or dispersion (refers to a distribution's extent of
stretching or squeezing) between values in a set of data. The lower the
standard deviation, the closer the data points tend to be to the mean (or
expected value), μ. Conversely, a higher standard deviation
indicates a wider range of values. Similar to other mathematical and
statistical concepts, there are many different situations in which standard
deviation can be used, and thus many different equations. In addition to
expressing population variability, the standard deviation is also often used to
measure statistical results such as the margin of error. When used in this
manner, standard deviation is often called the standard error of the mean, or
standard error of the estimate with regard to a mean. The calculator above
computes population standard deviation and sample standard deviation, as well
as confidence interval approximations.
Population Standard Deviation
The population standard deviation, the standard definition of σ,
is used when an entire population can be measured, and is the square root of
the variance of a given data set. In cases where every member of a population
can be sampled, the following equation can be used to find the standard
deviation of the entire population:
|
Where xi is an individual value |
For those unfamiliar with summation notation, the equation above
may seem daunting, but when addressed through its individual components, this
summation is not particularly complicated. The i=1 in the
summation indicates the starting index, i.e. for the data set 1, 3, 4, 7,
8, i=1 would be 1, i=2 would be 3, and so on.
Hence the summation notation simply means to perform the operation of (xi -
μ)2 on each value through N, which in this case
is 5 since there are 5 values in this data set.
EX: μ = (1+3+4+7+8) / 5 = 4.6
σ = √[(1 - 4.6)2 + (3 - 4.6)2 + ... + (8 - 4.6)2)]/5
σ = √(12.96 + 2.56 + 0.36 + 5.76 + 11.56)/5 = 2.577
Q.2: a) What is the difference between
precision and accuracy of a result? Explain with Examples.
What is Accuracy?
Accuracy is defined as ‘the degree to
which the result of a measurement conforms to the correct value or a standard’
and essentially refers to how close a measurement is to its agreed value.
What is Precision?
Precision is defined as ‘the quality of
being exact’ and refers to how close two or more measurements are to each
other, regardless of whether those measurements are accurate or not. It is
possible for precision measurements to not be accurate.
What is the difference between Accuracy
and Precision?
Both accuracy and precision reflect how
close a measurement is to an actual value, but they are not the same. Accuracy
reflects how close a measurement is to a known or accepted value, while
precision reflects how reproducible measurements are, even if they are far from
the accepted value. Measurements that are both precise and accurate are
repeatable and very close to true values.
Example of the difference between
Accuracy and Precision…
The example of a darts board is often used
when talking about the difference between accuracy and precision.
Accurately hitting the target means you
are close to the centre of the target, even if all the marks are on different
sides of the centre. Precisely hitting a target means all the hits are closely
spaced, even if they are very far from the centre of the target.
How can Precisa help you with your
Precision Measurements?
At Precisa, ensuring the precision of
your measurements is our top priority.
Not only do we produce analytical and precision
balances which measure up to 0.1 mg, but we also offer calibration services
across the UK which guarantee that your readings are 100% precise and accurate.
b) What are two broad categories of
errors in data collected by sample surveys? What are the methods for reducing
sampling error?
The accuracy of a survey estimate refers
to the closeness of the estimate to the true population value. Where there is a
discrepancy between the value of the survey estimate and true population value,
the difference between the two is referred to as the error of the survey
estimate. The total error of the survey estimate results from the two types of
error:
sampling error, which arises when only a
part of the population is used to represent the whole population; and
non-sampling error which can occur at any
stage of a sample survey and can also occur with censuses. Sampling error can
be measured mathematically whereas measuring non-sampling error can be
difficult.
It is important for a researcher to be
aware of these errors, in particular non-sampling error, so that they can be
either minimised or eliminated from the survey. An introduction to measuring
sampling error and the effects of non-sampling error is provided in the
following sections.
The target population may not be clearly
defined through the use of imprecise definitions or concepts. The survey
population may not reflect the target population due to an inadequate sampling
frame and poor coverage rules. Problems with the frame include missing units,
deaths, out-of-scope units and duplicates. These are discussed in detail in
Frames and Population.
Non-Response Bias
Non-respondents may differ from
respondents in relation to the attributes/variables being measured.
Non-response can be total (none of the questions answered) or partial (some
questions may be unanswered owing to memory problems, inability to answer,
etc.). To improve response rates, care should be taken in designing the
questionnaires, training of interviewers, assuring the respondent of confidentiality,
motivating him/her to co-operate, and calling back at different times if having
difficulties contacting the respondent. "Call-backs" are successful
in reducing non-response but can be expensive for personal interviews.
Non-response is covered in more detail in Non-Response.
Questionnaire problems
The content and wording of the
questionnaire may be misleading and the layout of the questionnaire may make it
difficult to accurately record responses. Questions should not be loaded,
double-barrelled, misleading or ambiguous, and should be directly relevant to
the objectives of the survey.
It is essential that questionnaires are
tested on a sample of respondents before they are finalised to identify
questionnaire flow and question wording problems, and allow sufficient time for
improvements to be made to the questionnaire. The questionnaire should then be
re-tested to ensure changes made do not introduce other problems. This is
discussed in more detail in Questionnaire Design.
Respondent Bias
Refusals to answer questions, memory
biases and inaccurate information because respondents believe they are
protecting their personal interest and integrity may lead to a bias in the
estimates. The way the respondent interprets the questionnaire and the wording
of the answer the respondent gives can also cause inaccuracies. When designing
the survey you should remember that uppermost in the respondent's mind will be
protecting their own personal privacy, integrity and interests. Careful
questionnaire design and effective questionnaire testing can overcome these
problems to some extent.
Respondent bias is covered in more detail
below.
Processing Errors
There are four stages in the processing
of the data where errors may occur: data grooming, data capture, editing and
estimation. Data grooming involves preliminary checking before entering the
data onto the processing system in the capture stage. Inadequate checking and
quality management at this stage can introduce data loss (where data is not
entered into the system) and data duplication (where the same data is entered
into the system more than once). Inappropriate edit checks and inaccurate
weights in the estimation procedure can also introduce errors to the data. To
minimise these errors, processing staff should be given adequate training and
realistic workloads.
Misinterpretation of Results
This can occur if the researcher is not
aware of certain factors that influence the characteristics under
investigation. A researcher or any other user not involved in the collection stage
of the data gathering may be unaware of trends built into the data due to the
nature of the collection, such as it's scope. (eg. a survey which collected
income as a data item with the survey coverage and scope of all adult persons
(ie. 18 years or older), would expect to produce a different estimate than that
produced by the ABS Survey of Average Weekly Earnings (AWE) simply because AWE
includes persons of age 16 and 17 years as part of it's scope). Researchers
should carefully investigate the methodology used in any given survey.
Time Period Bias
This occurs when a survey is conducted
during an unrepresentative time period. For example, if a survey aims to
collect details on ice-cream sales, but only collects a weeks worth of data
during the hottest part of summer, it is unlikely to represent the average
weekly sales of ice-cream for the year.
c) A random sample of 36 cases is drawn
from a negatively skewed probability distribution with mean of 2 and standard
deviation of 3. Find the mean and standard error of the sampling distribution
of X (6+7+7)
First we have to find the mean of the given
data;
Mean = (5+10+12+15+20)/5 = 62/5 = 10.5
Now, the standard deviation can be calculated
as;
S = Summation of difference between each value
of given data and the mean value/Number of values.
Hence,
After solving the above equation, we get;
S = 5.35
Therefore, SE can be estimated with the
formula;
SE = S/√n
SE = 5.35/√5 = 2.39
Q.3: a) What
is meant by estimation? Distinguish between point estimate and interval
estimate. Why is an interval estimate more useful?
Estimation (or estimating) is the process
of finding an estimate or approximation, which is a value that is usable for
some purpose even if input data may be incomplete, uncertain, or unstable. The
value is nonetheless usable because it is derived from the best information
available.[1] Typically, estimation involves "using the value of a
statistic derived from a sample to estimate the value of a corresponding
population parameter".[2] The sample provides information that can be
projected, through various formal or informal processes, to determine a range
most likely to describe the missing information. An estimate that turns out to
be incorrect will be an overestimate if the estimate exceeds the actual
result[3] and an underestimate if the estimate falls short of the actual
result.
Point Estimation
Anna surveys 100 random people out of her
town about building a dog park. This is known as a sample. A sample is a part
of a population used to describe the whole group. Anna will use the data that
she gathers from this sample to describe the population of the town. The data
Anna gathers will be statistics.
A statistic is the characteristics of a
sample used to infer information about the population. For example, Anna may
include questions on her survey such as age and number of pets. If the mean age
for her sample is 32, then 32 is a sample statistic. She might infer that the
average age of the people in town is also 32.
A point estimation is a type of
estimation that uses a single value, a sample statistic, to infer information
about the population. Point estimation can be a sample statistic. The sample
mean of age for the sample, 32, can be used as a point estimation.
Point estimation is a single value that
can be inferred as a population parameter. Let's discuss populations,
parameters, and their relationship to point and interval estimations.
Interval Estimation
After Anna collected her data from the
survey, she can now draw inferences from her sample statistics about the
population. A population is all members of a specified group. In this case, the
population is literally the population of a town. However, a population can
mean many things. For example, if you are doing research about college
students, then your population would be all college students. If your research
is about high school athletes, then your population would be all high school
athletes. It's mostly up to the researcher to define the population.
Once sample information is gathered, the
researcher can make inferences about the population, known as a parameter. A
parameter is the characteristics used to describe a population. You can use a
sample statistic to develop population parameters. For example, Anna found that
of the 100 people she surveyed, each household owns an average of two pets.
This is a sample statistic. However, Anna can also use this statistic to infer
that everyone in the population also owns an average of two pets; at this point
it is considered a population parameter. If we were estimating from a single
value, such as the average of two pets, then we could say that this population
parameter was a point estimation.
However, what about the people that own
more than two pets or no pets? Can 100 people really show us the
characteristics of an entire town? This is where interval estimation comes in
handy. Interval estimation is the range of numbers in which a population
parameter lies considering margin of error. Because there is a certain level of
uncertainty, an interval estimate gives a range, rather than a single value, of
the population parameters.
b) A
poll is taken among the residents of a city and the surrounding country to determine
the feasibility of a proposal to construct a civic center. If 2400 of 5000 city
residents favor the proposal and 1200 of 2000 country residents favor it, find
a 95% confidence interval for the true difference in the proportions favouring
the proposal to construct the civic center. (10+10)
Q.4: a) Why
is the z-test usually inappropriate as a test statistic when sample size is
small?
There’s a reason Z Test is a necessity
for small sample size: it is not just a test for the importance of a specific
test (which you should use), but it is an umbrella term for the quantity of
information that can be found in a single document. It’s not just that you need
to have an independent test, but that you need an independent test as well. The
Z Test is an umbrella word for the quantity and quality of information that may
be found in the standard document. It is rather a general term that exists in
many other fields and is used to refer to the quantity of data that can be seen
in a single or multiple documents. For example, the following document may be
considered an example of a document that contains information about the number
of items that you will be asked to create in the future: The examples below are
just two examples that use an independent test. You may want to check out the Z
Test on the main page, as it is a good tool for identifying what is expected of
small sample size. Just as an example, the Z Test will be applied as follows: Z
Test is inappropriate for small sample sizes. 1. Your name and your address
Since the Z Test is the standard test for small sample and not just for the
number of samples, it is necessary to include a name or address of the project
that contains your name or address. If you have a name or an address like: John
Smith You may be asked to name your project Smith. The Z Test will give you the
name you filled in. Z test is inappropriate. 2. Your name If you are asked to
name a project Smith, you will be called Smith. If you are asked who Smith is,
you will name Smith. The test will be: name Smith Your name Smith is Smith. 3.
Your project name or address A project name is a numerical value that is used
to represent the project name of a project. For example, you will have to have
a project name: Project Smith Project Name: John Smith The test will be name
John Smith – Name click to find out more – – – 5. Your project title or project
name The name you will name the project will be: Project Smith Name: John Smith
– Project Name: John The project title or title may be a number or a string.
b)
A manufacturer of house dresses sent out advertising by mail. He sent
samples of material to each of two groups of 1000 women. For one group he
enclosed white returns envelop and for the other group, a blue envelop, he
received orders from 9% and 12% respectively. Is it quite certain that the blue
envelop will help sals.
Use
a=0.05. (10+10)
Hypothesis:
Given that,
n1 = 1000 p1 =
10% = 0.10
n2 = 1000 p2 =
13% = 0.13
Pooled
proportion :
Test statistic,
P-value = 2*P(z
< -2.103) = 2* 0.0179 = 0.0358
Decision rule :
If p-value is less than level of significance then reject H0, otherwise accept
it.
Here p-value is
less than 0.05, hence reject H0 .
The proportion
of orders received is different at 5% level of significance. Hence we conclude
that the colour of the envelope has an effect on the sale.
Q.5: a) An
electric company claimed that at least 85% of the parts which they supplied
conformed to specifications. A sample 400 parts was tested and 75 did not meet
specifications. Can we accept the company’ claim at 0.05 level of significance?
Ho: p >= 0.95 (claim)
Ha: p < 0.95
-----
p-hat = 182/200 = 0.91
z(0.41) = (0.91-0.95)/sqrt[0.95*0.05/200]
= -0.04/sqrt[0.95*0.05/200]
= -2.5955
--------------
p-value = P(z < -2.5955) = 0.0047
------
Conclusion: Since the p-value is less
than 1%, reject Ho ; reject the claim.
b) In
a random sample of 1000 house in a certain city, 618 own color TV sets. Is this
sufficient evidence to conclude that 2/3 of houses in this city have color TV
sets? Use a +0.02. (10+10)
Null hypothesis
is:
H0 :
p < 2/3
Alternative
hypothesis is:
H1 :
p = 2/3
population
proportion: p = 2/3 = 0.667
sample size: n=
1000, x= 618
sample
proportion : = x/ n
=618 / 1000 = 0.618
formula for
test statistics is
z= - 3.29
test statistics
= -3.29
So the p value
at 0.02 significance is = 0.005
So it is less
than 0.05, hence we reject the null hypothesis.
SO the
alternative hypthesis is true i.e. there is sufficient evidence.
Dear Student,
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