Elements of inferential statistics

(Basic concepts, Type I and Type II error, p-value, power, one tailed and
two tailed test and confidence interval)
Prepared by
Arati G Mishra
PhD Scholar
Principal
SET SON-Junagadh

Inferential statistics
Estimation of
Parameters
Hypothesis Testing
Point
Estimation
Interval
Estimation
Parametric
Tests
Non Parametric
Tests

 Inferential statistics is the act of generalizing from the data (“sample”)
to a larger phenomenon (“population”) with calculated degree of
certainty.
 The act of generalizing and deriving statistical judgments is the process
of inference.
 Inferential statistics is a statistical method that deduces from a small but
representative sample the characteristics of a bigger population.
 In other words, it allows the researcher to make assumptions about a
wider group, using a smaller portion of that group as a guideline.
 Inferential Statistics makes inference and prediction about population
based on a sample of data taken from population.
 It is simply used for explaining meaning of descriptive statistics.

 To determine difference between experimental and control group in
experimental research.
 To enable researcher to evaluate effects of an independent variable on
dependent variable.
 To determine whether the findings from the sample can generalize - or
be applied to the entire population
 To estimate differences in scores between groups in a research study.

 The two forms of inferential statistics.
• Estimation
• Hypothesis testing
 There are two forms of estimation:
• Point estimation (maximally likely value for parameter)
• Interval estimation (also called confidence interval for parameter)
 Point estimates are single points that are used to infer parameters
directly.
 For Example, Sample proportion pˆ (“p hat”) is the point estimator of p
 Sample mean x (“x bar”) is the point estimator of μ
 Sample standard deviation s is the point estimator of σ

 Point Estimate: A single statistic value that is the “best guess” for
the parameter value.
 Interval Estimate: An interval of numbers around the point
estimate, that has a fixed “confidence level” of containing the
parameter value, Called a confidence interval confidence interval.
 Point estimates and parameters represent fundamentally different
things.
• Point estimates are calculated from the data; parameters are not.
• Point estimates vary from study to study; parameters do not.
• Point estimates are random variables: parameters are constants.

 A point estimate of the population mean almost is assured of being
in error, the estimate from the sample will not equal to the exact
value of the parameter.
 To gain confidence about the accuracy of this estimate we may also
construct an interval of scores that is expected to include the value
of the population mean. Such intervals are called confidence
interval.
• A confidence interval is a range of scores that is expected to
contain the value of (μ). The lower and upper scores that determine
the interval are called confidence limits.
• A level of confidence can be attached to this estimate so that the
researcher can be 95% or 99% confidence level that encompasses
the population mean.

 Unbiased: Sampling distribution of the estimator centers around the
parameter value.
e.g. Biased estimator: sample range. It cannot be larger than
population range.
 Efficient: Smallest possible standard error compared : Smallest
possible standard error, compared to other estimators
e.g. If population is symmetric and approximately normal in shape,
sample mean is more efficient than sample median in estimating the
population mean and median.

 A confidence interval is the mean of estimate plus and minus the
variation in that estimate.
 It is the range of values expected to estimate within a certain level
of confidence.
 Confidence, in statistics, is another way to describe probability.
 A confidence interval displays the probability that a parameter will
fall between a pair of values around the mean.
 Confidence intervals measure the degree of uncertainty or
certainty in a sampling method.
 They are most often constructed using confidence levels of 95% or
99%.

 For example, if you construct a confidence interval with a 95%
confidence level, you are confident that 95 out of 100 times the
estimate will fall between the upper and lower values specified by
the confidence interval.
 Confidence level is usually one minus the alpha ( a ) value.
 Confidence level = 1 − a
 So if we use an alpha value 5% then ci would be 95%.

 Hypothesis testing is a type of inferential procedure that takes help
of sample data to evaluate and assess credibility of a hypothesis
about a population.
 Inferential statistics are generally used to determine how strong
relationship is within sample. But it is very difficult to obtain a
population list and draw a random sample.
 These tests are performed under two main hypotheses. They are,
Null hypothesis and Alternative hypothesis.
 For Example the research prediction that is tested (e.g. students in
situation A will perform better than students in situation B) Null
hypothesis: a statement of “no difference” between the means of
two populations (there will be no difference in the performance of
students in situations A and B).

 Null Hypothesis: the hypothesis that sample observations result
purely from chance. The null hypothesis tends to state that there’s
no change.
 Alternative Hypothesis: the hypothesis that sample observations are
influenced by some non-random cause.
 P-value: the probability of obtaining the observed results of a test,
assuming that the null hypothesis is correct; a smaller p-value means
that there is stronger evidence in favor of the alternative hypothesis.
 Alpha: the significance level; the probability of rejecting the null
hypothesis when it is true — also known as Type 1 error.

 State null and alternative hypotheses.
 Set significance level, the alpha. This is typically set at 5% but can
be set at other levels depending on the situation and how severe it
is to committing a type 1 and/or 2 error.
 Collect sample data and calculate sample statistics.
 Calculate the p-value given sample statistics. Once you get the
sample statistics, you can determine the p-value through different
methods
 Reject or do not reject the null hypothesis.

 A type I error is a kind of fault that occurs during the hypothesis testing
process when a null hypothesis is rejected, when it’s actually true.(even
though it is accurate and should not be rejected).
 In hypothesis testing, a null hypothesis is established before the onset of
a test. These false positives are called type I errors.
 The risk of committing this error is the significance level (alpha or α) we
choose. That’s a value that we set at the beginning of our study to assess
the statistical probability of obtaining your results (p value).
 If the p value of our test is lower than the significance level, it means our
results are statistically significant and consistent with the alternative
hypothesis. If your p value is higher than the significance level, then our
results are considered statistically non-significant.

 A Type II error means not rejecting the null hypothesis when it’s actually
false.
 This is not quite the same as “accepting” the null hypothesis, because
hypothesis testing can only tell you whether to reject the null
hypothesis.
 Instead, a Type II error means failing to conclude there was an effect
when there actually was.
 A type II error is a statistical term used within the context of hypothesis
testing that describes the error that occurs when one accepts a null
hypothesis that is actually false.
 A type II error produces a false negative, also known as an error of
omission.

 One-tailed test alludes to the significance test in which the region
of rejection appears on one end of the sampling distribution.
 It represents that the estimated test parameter is greater or less
than the critical value.
 When the sample tested falls in the region of rejection, i.e. either
left or right side, as the case may be, it leads to the acceptance of
alternative hypothesis rather than the null hypothesis.
 It is primarily applied in chi-square distribution; that ascertains the
goodness of fit.
 In this statistical hypothesis test, all the critical region, related
to α, is placed in any one of the two tails. One-tailed test can be:

 Left-tailed test: When the population parameter is believed to be
lower than the assumed one, the hypothesis test carried out is the
left-tailed test.
 Right-tailed test: When the population parameter is supposed to be
greater than the assumed one, the statistical test conducted is a
right-tailed test.

 The two-tailed test is described as a hypothesis test, in which the
region of rejection or say the critical area is on both the ends of
the normal distribution.
 It determines whether the sample tested falls within or outside a
certain range of values. Therefore, an alternative hypothesis is
accepted in place of the null hypothesis.
 In this test, α is bifurcated into two equal parts, placing half on
each side, i.e. it considers the possibility of both positive and
negative effects.
 It is performed to see, whether the estimated parameter is either
above or below the assumed parameter, so the extreme values,
work as evidence against the null hypothesis.

BASIS OF COMPARISON ONE-TAILED TEST TWO-TAILED TEST
Meaning A statistical hypothesis test in which
alternative hypothesis has only one
end, is known as one tailed test.
A significance test in which
alternative hypothesis has two ends,
is called two-tailed test.
Hypothesis Directional Non-directional
Region of rejection Either left or right Both left and right
Determines If there is a relationship between
variables in single direction.
If there is a relationship between
variables in either direction.
Result Greater or less than certain value. Greater or less than certain range of
values.
Sign in alternative > or < ≠

 Power refers to the probability that your test will find a statistically
significant difference when such a difference actually exists.
 Power analysis is normally conducted before the data collection. The
main purpose underlying power analysis is to help the researcher to
determine the smallest sample size that is suitable to detect the effect
of a given test at the desired level of significance.
 Power is a function of three primary factors and one secondary factor:
sample size, effect size, significance level, and the power of the statistic
used. The most common reason to conduct a power analysis is to
determine the sample size needed for a particular study.
 Power is the extent to which a test can correctly detect a real effect
when there is one. A power level of 80% or higher is usually considered
acceptable.

 The level of significance is defined as the fixed probability of wrong
elimination of null hypothesis when in fact, it is true, whether the
null hypothesis is assumed to be accepted or rejected.
 The level of significance is stated to be the probability of type I
error and is preset by the researcher with the outcomes of error.
The level of significance is the measurement of the statistical
significance.
 Significance Level = p (type I error) = α
 Levels of Significance is used to indicate the chance that we are
wrong in rejecting the null hypothesis and also called the level of
probability or p level p=.01, for example, means that the probability
of finding the stated difference as a result of chance is only 1 in 100

 p-value stands for probability value.
 Under probability distribution, once a value is identified on the
probability distribution, then the area left on either side is known as the
probability.
 If the value is close to the mean, it would have a large amount of area
under the curve, i.e., a high p-value, which indicates that it is a common
value.
 The probability of finding such valuable data is also high. On the contrary,
if the value is far away from the mean, then the p-value will be low,
indicating the probability of finding such a value in the data is low.
 This concept is used in hypothesis testing to accept or reject the null
hypothesis.

 To define if the p-value is high or not, we decide an alpha value.
If the p-value is higher than the alpha value, then we consider the
p-value to be high, indicating that the value is commonly found in
the data.
 Hence, it is not statistically significantly different from the mean
allowing us to accept the Null Hypothesis.

 The alpha value is the probability threshold for statistical significance.
 The most common alpha value is p = 0.05, but 0.1, 0.01, and even 0.001
are sometimes used .
 P-Value vs. Alpha: What’s the Difference?
 Two terms that we often get confused in statistics are p-
value and alpha.
 Both terms are used in hypothesis tests, which are formal statistical tests
we use to reject or fail to reject some hypothesis.
 For example, suppose we hypothesize that a new pill reduces blood
pressure in patients more than the current standard pill.

 To test this, we can conduct a hypothesis test where we define the
following null and alternative hypotheses:
 Null hypothesis: There is no difference between the new pill and
the standard pill.
 Alternative hypothesis: There is a difference between the new pill
and the standard pill.
 If we assume the null hypothesis is true, the p-value of the test
tells us the probability of obtaining an effect at least as large as
the one we actually observed in the sample data.
 For example, suppose we find that the p-value of the hypothesis
test is 0.02.

 Here’s how to interpret this p-value: If there truly was no
difference between the new pill and the standard pill, then 2% of
the times that we perform this hypothesis test we would obtain the
effect observed in the sample data, or larger, simply due to random
sample error.
 This tells us that obtaining the sample data that we actually did
would be pretty rare if indeed there was no difference between the
new pill and the standard pill. Thus, we would be inclined to reject
the statement in the null hypothesis and conclude that there is a
difference between the new pill and the standard pill.
 But what threshold should we use to determine if our p-value is low
enough to reject the null hypothesis?
 This is where alpha comes in.

 The Alpha Level
 The alpha level of a hypothesis test is the threshold we use to
determine whether or not our p-value is low enough to reject the
null hypothesis. It is often set at 0.05 but it is sometimes set as low
as 0.01 or as high as 0.10.
 For example, if we set the alpha level of a hypothesis test at 0.05
and we get a p-value of 0.02, then we would reject the null
hypothesis since the p-value is less than the alpha level.
 Thus, we would conclude that we have sufficient evidence to say
the alternative hypothesis is true.
 It’s important to note that the alpha level also defines the
probability of incorrectly rejecting a true null hypothesis.

 As mentioned earlier, the most common choice for the alpha level of a
hypothesis test is 0.05. However, in some situations where there are
serious consequences for making incorrect conclusions, we may set the
alpha level to be even lower, perhaps at 0.01.
 For example, in the medical field it’s common for researchers to set the
alpha level at 0.01 because they want to be highly confident that the
results of a hypothesis test are reliable.
 Conversely, in fields like marketing it may be more common to set the
alpha level at a higher level like 0.10 because the consequences for
being wrong aren’t life or death.
 It’s worth noting that increasing the alpha level of a test will increase
the chances of finding a significance test result, but it also increases the
chances that we incorrect.
How to Choose the Alpha
Level ?

1. A p-value tells us the probability of obtaining an effect at least as
large as the one we actually observed in the sample data.
2. An alpha level is the probability of incorrectly rejecting a true null
hypothesis.
3. If the p-value of a hypothesis test is less than the alpha level, then
we can reject the null hypothesis.
4. Increasing the alpha level of a test increases the chances that we
can find a significant test result, but it also increases the chances
that we incorrectly reject a true null hypothesis.

 https://www.statisticshowto.com/probability-and-statistics/statistics-definitions/inferential-
statistics/
 https://libraryguides.centennialcollege.ca/c.php?g=717168&p=5123291
 https://slideplayer.com/slide/6261841/
 https://www.intellspot.com/inferential-statistics-types/
 https://www.geneseo.edu/~rommel/educ504/ch10%20inferential%20stx.pdf
 https://www.geeksforgeeks.org/introduction-of-statistics-and-its-types/IMP
 https://askinglot.com/what-is-the-purpose-of-inferential-statistics
 https://www.sjsu.edu/faculty/gerstman/StatPrimer/estimation.pdf
 http://users.stat.ufl.edu/~aa/harvard/5.%20Estimation.pdf
 https://www.statology.org/p-value-vs-alpha/
 https://www.statsdirect.com/help/basics/p_values.htm
 https://towardsdatascience.com/hypothesis-testing-explained-as-simply-as-possible-6e0a256293cf
 https://www.scribbr.com/statistics/type-i-and-type-ii-errors/

Elements of inferential statistics

Elements of inferential statistics

Elements of inferential statistics

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Elements of inferential statistics