POINT ESTIMATE, THE STANDARD DEVIATION AND THE HYPOTHESIS
TESTING
Estimation represents ways or a process of learning and determining
the population parameter based on the model fitted to the data. There are two
types of estimates – point estimates and confidence interval estimates.
Confidence Intervals gives a range of values for the parameter interval
estimates are intervals within which the parameter is expected to fall, with a
certain degree of confidence. The level of confidence is denoted by 1 – alpha,
and is called the confidence level of the intervals and determines the
probability of rejection (acceptance) of the null hypothesis, which is actually
true, the level of reliability of the test. Point estimation involves the use of
sample data to calculate a single value, known as a point estimate since it
identifies a point in some parameter space, which is to serve as a "best guess"
or "best estimate" of an unknown population parameter. The standard
deviation is a statistic that measures the dispersion of a dataset relative to its
mean and is calculated as the square root of the variance. Standard deviation
is a statistical measure that shows the spread of the most used data for the
quantitative scale numbers in general. If many data are close to the average,
the standard deviation is small; if many data are spread far from the mean,
the standard deviation is large. If all data values are exactly the same, the
standard deviation value is zero. Hypothesis testing is a form of statistical
inference that uses data from a sample to draw conclusions about a
population parameter or a population probability distribution. One
hypotheses is called the null hypothesis (statistical hypothesis) and the other
is called Alternative Hypothesis (research hypothesis). The zero hypothesis is
a general proposition that accepts that there is no situation beyond
expectations, for example, there is no relationship between groups or
variables, or that there is no difference between the two cases measured. The
alternative hypothesis is the hypothesis that should be accepted in case of
rejection of the null hypothesis.
Point estimation, standard deviation and hypothesis testing are related
terms. We can explain the relationship between point estimation and standard
deviation as follows: point estimation gives exact results, but standard
deviation shows its spread by mean. Therefore, there is no similarity
relationship between them. However, it can be said that the standard deviation
has a similarity with the hypothesis test. Because both of them give results in
a range. And the values are estimated values. They do not give exact results
as in point estimation.
After examining the relationship between them, I will give examples
about point estimation, the standard deviation and the hypothesis tests.
Firstly, I will give example about point estimation. For example we wanted to
estimate the average time that 12-year-olds should run 100 yards. The average
working time of a random sample of 12-year-olds would be an estimate of the
average working time of 12-year-olds. Thus, the sample mean, M, would be a
point estimate of the population mean, μ. Second example is about the
standard deviation. Each of the three groups {0, 0, 14, 14}, {0, 6, 8, 14} and
{6, 6, 8, 8} has an average (mean) of 7. But their standard deviations are 7, 5,
and 1. The third group has a much smaller standard deviation than the other
two because its numbers are all close to 7. The basic idea is, the standard
deviation tells us how far from the average the rest of the numbers tend to be.
It will have the same units as the numbers themselves. If, for example, the
group {0, 6, 8, 14} is the ages of a group of four brothers in years, the average
is 7 years and the standard deviation is 5 years. Final example is about the
hypothesis test. Milk is a nutritional source that helps bone development,
especially for children.
1- Null hypothesis - Children who do not drink milk are no less likely to
become ill because of undevelopment bone.
2- Alternative hypothesis - Children who do not drink milk are less likely
to become ill because of undevelopment bone.
If we need to give an example to explain these three;
It is wondered whether the height of the girls in a school is above the
world average. The average height of their daughters in the world is 160 cm.
The height of the randomly selected girls in the school is 159, 160, 166, 170,
163 and 1,62 with α(alpha) =0,05
H0: μ =160 => The height of girls in school are on average or below. => μ<= 160
H1: μ #160 => The height of girls in school are above average. => μ> 160

Firstly, the average height of 6 girls should be calculated.
159+160+166+170+163+162= 980/6 =163,3 => mean of sample

Secondly, the variance must be ound for standard deviation.
(159-163,3)2+ (160-163,3)2+(166-163,3)2+(170-163,3)2
163,3)2
=18,49+10,89+7,29+77,89+0,9+1,69=
+
(163-163,3)2+ (162-
117,15
117,15/6= 19,525 => variance(σ2)
The standard deviation is √19.525 =4,41 = σ
Z Test
=
(x̄ – μ) / (σ / √n)
Z Test
=
(163,3-160)/(4,41/ √6) =1,83


The calculated z= [1,83] value is above the z value in the table.
Confidence Interval: 163,3 ± (1,96 × 4,41/ √6 )= 159,78…………166.82