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Running head: HYPOTHESIS TESTING
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Hypothesis Testing
Name
Institution
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Introduction
A confidence interval is basically the range of the values which are derived from the
sample statistics which is likely to encompass the values of unknown parameters of the
population. Due to the random nature of the population, it is likely that they will yield same
confidence interval. However, when a statistician repeats a given sample of data for several
times, there is high chance that a certain percentage of the outcome in the confidence interval
would have indefinite parameters of the population. Thus, the confidence intervals which contain
the factors are the interval of the confidence interval. Mostly, the confidence intervals are used to
bind the average or the mean as well as the standard deviation. On the other hand, it can be used
in obtaining the regression coefficients, the proportions of the occurrences and to note the
differences of a given data. A hypothesis is mainly s proposed for any two data sets. In this case,
it gives both the null hypothesis and the alternative hypothesis which gives the relationship
between two data sets which will aid in determining the level of significance. To do so, I
decided to sample data of a given production firm. The aim of doing so was to determine if the
renewal of the machine in the production process improves the performance. I sampled data
about amount of time the old machine takes to pack ten cartons. I collected the data randomly on
different days chosen randomly within the firm. It was important to collect the data as it will aid
in determining whether the firm needs to buy the new machines to improve its production
process.
Process of Collecting Data
To collect the data it was necessary to follow some of the procedures I set down. I
decided that it was good to use observation and interviewing method to collect the data. After,
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seeking permission with the authorities of the firm, I observed and recorded the time taken by the
old machine to pack ten cartons with the use of stop watch. After, the purchase of then new
machine I went and observed as I determined the time taken by the new machine to pack ten
cartons of the jars. I also collected several data for different days which I chose randomly. I
collected the data below as displayed in figure 1 and then plotted a bar graph to represent the two
data as shown in figure 2.
Figure 1
New
42.1
41.3
42.4
43.2
41.8
41.8
41.0
42.8
41.3
42.7
42.7
43.8
42.5
43.1
44.0
43.3
43.6
43.5
41.7
44.1
Machine
Old
Machine
The bar of the above data is as shown below in the figure 2.
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Figure 2
44.5
44
43.5
43
42.5
42
New
Old
41.5
41
40.5
40
39.5
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Focus
Comparison of the Packing Machines
In the firm, the machine packs the cartons jars in the cartons. It was stipulated by the
manager that the new machine will pack faster than the old machine. To taste the hypothesis, I
used the table of the data as shown in figure 1. I formulated hypothesis concerning the data I
collected.
I wanted to determine if the data I collected provided enough evidence to conclude that
on an average basis a new machine packs faster than the old machine. I performed the hypothesis
test at a 5% significance level. Therefore the data is given as follows:
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Ӯ1= 42.14, s1=0.683
Ӯ2= 43.23,
s2=0.750
First Assumption
It is always good to ensure that the samples are independent. The data given above are
independent since they are from two machines.
Second Assumption
Are the samples provided above large or mainly normal population? In this data sample
we have n1<30 and the n2<30.Thus, we do not have enough samples to work with, which means
that it is necessary to check the normality assumption of the two data (Milton, 2009). Thus, it is
necessary to plot a normality plots for the data. They are as shown in Figure 3 and figure 4.
Figure 3
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Figure 4
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As the two plots display a normal distribution, it is right for to conclude that the data
come from normal distributions.
Third Assumption
It is important to determine if the two populations have equal variance. According to the
calculations, s1 and s2 are not so much different. I concluded this using the rule of the thumbs
where by the ratio of two sample data standard deviation is from 0.5 to 2. They are not that
different. It is because s1/s2 = 0.683/0.750= 0.91. The result is close to 1. Let’s continue the
pooled t- test. Let’s use μ1 to denote the mean of the new machine while μ2 to represent the
mean of the old one.
Step One
H0: μ1−μ2 =0H
Ha: μ1−μ2< 0
Step Two
Significance level
α=0.05
Step Three
Computation of the t-statistic
sp = √{9 × (0.683)2 + 9 × ( 0.750)2 } ÷ {10 + 10 − 2} = 0.717
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t= 42.14 - 43.23/ {0.717*√ (1/10 +1/10)} = -3.40
Step Four
Left-tailed test
Critical value = −tα = −t0.05
Degrees of freedom = 10+10−2 =18
−t 0.05= −1.734 Thus; Rejection region t∗<−1.734
Step Five
It is also important to check if the test statistic falls inside the rejection region or not so that we
can decide whether to reject Ho or not.
t∗=−3.40<−1.734
Reject Ho at α=0.05
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Step Six
At this point we are able to conclude the above hypothesis testing. At a significance of
5% the data has provided us with evidence that the new machine packs at a faster rate than the
old machine averagely.
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References
Milton, M. (2009). Head first data analysis. Beijing: O'Reilly.