Hypothesis Testing
We have methods to test our hypothesis and these methods can
be categorized into two parts.
Parametric Testing: This type of tests make
assumptions about the Population parameters and the distributions that the data
came from. These types of test includes Student's T tests and ANOVA tests, which assume data
is from a normal distribution.
Non- parametric Testing: Non-parametric tests are used when there is no or few
information available about the population parameters.
Z Test:
To find
test statistics, we can use the below formula.
Z test can be done if the
below 3 points are satisfied.
1.
Sample size should be
> 30.
2.
Population SD should be
known.
3.
Variables should be
continues.
Steps for Z Test:
1.
State Null &
Alternate Hypothesis.
2.
Find the Level of
significance (α).
3.
Find Critical Values.
4.
Find test statistic.
5.
Make conclusion.
Example:
1,500 women followed the Atkin’s diet for a month. A random sample of 29 women gained an average of 6.7 pounds. Test the hypothesis that the average weight gain per woman for the month was over 5 pounds. The standard deviation for all women in the group was 7.1.
1,500 women followed the Atkin’s diet for a month. A random sample of 29 women gained an average of 6.7 pounds. Test the hypothesis that the average weight gain per woman for the month was over 5 pounds. The standard deviation for all women in the group was 7.1.
Given:
Population: 1500
Population SD: 7.1
Sample Size: 29
Sample Mean: 6.7 pounds
To test: Test the hypothesis that the
average weight gain per woman for the month was over 5 pounds.
So here the average weight gain of the
total population is 5 pounds. So population mean is 5 pounds. We need to test
the hypothesis that average weight is above 5 pounds.
Obviously our Null hypothesis is µ >
5 pounds.
If we reject the null hypothesis then
it means the sample average weight gained is close to or less than the
population average weight gained.
Step #1:
i.e. H0: µ > 5
As H0 > 5, our alternate hypothesis
is the opposite of Null Hypothesis.
Here H1 <= 5
Step #2: Level of
Significance
As the confidence/significance level is
not given then we can consider it as 95% confidence level.
C = 0.95
α = 1 – 0.95 = 0.05
Step #3: Critical Value for 95% confidence level α is 1.645.
Step #4:
The Sample mean – population mean = 6.7
– 5 = 1.7
σ /√n = 7.1/√29 = 7.1/5.38 = 1.3197
z = 1.289 which is < 1.645
Step #5: Conclusion
As the z value is less than critical
value, the z value falls in accept region.
So we accept the Null Hypothesis.
T Test
or Students T Test:
If the sample size is <30 i.e.
n<30 then those samples are considered as small samples.
The methods used for small samples can
be used with large samples. But the vice versa is not appropriate.
Per Central Limit Theorem, the
distribution will become normal when the sample size increases.
With a small sample the distribution
may not be normal, even if it looks normal it is not more like a bell curve as
a normal distribution.
When sample size is 30 or less than 30
and the population SD is unknown, then we can use the t-distribution.
Conventions of T Test:
1.
The population from which
the samples are drawn is normal
2.
Sample is random.
3.
Population SD is not
known.
Applications of t-test:
1.
Test of hypothesis about population
mean.
2.
Test the hypothesis about
the difference between two means.
3.
Test the hypothesis about
coefficient of correlation.
Formula:
To find
the interval: ẍ ± t-table value * (s/√n)
To test
the hypothesis: (ẍ - 𝛍)/(s/√n)
Example:
A company wants to improve sales. Past
sales data indicate that the average sale was $100 per transaction. After
training the company’s sales force, recent sales data (taken from a sample of
25 salesmen) indicates an average sale of $130, with a standard deviation of
$15. Did the training work? Test your hypothesis at a 5% alpha level.
Step #1:
H0: µ = 100
H1: µ > 100
Step #2:
Level of significance α = 0.05
Step #3:
x̄ = $130.
μ = $100
sample standard deviation(s) = $15.
Sample Size (n) = 25.
Find the t-table value.
Look up 24 degrees of freedom in the
left column and 0.05 in the top row. The intersection is 1.711.This is your
one-tailed critical t-value.
T value is 1.711.
Step #4:
(130 – 100) / (15/RootOf 25)
= 30/3 = 10
Step #5:
The calculated value is greater than T
value 1.711
Conclusion: We fail to
accept the null hypothesis.
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