Z test (Quantitative) is applied to compare two independent sample means, when sample size is more than 30 / population standard deviation is known.

Assumptions
1. Data is continuous and normally distributed in both the groups.
2. Samples are drawn using random sampling technique.
3. Population Standard deviation for both the groups are known. (If not, in case of large sample size (> 30): Sample SD can be taken as a measure of population SD.)

Steps:

1. Null hypothesis and Alternate Hypothesis

Null Hypothesis

A. Two tailed test: Two independent sample means are equal. OR Sample means are not significantly different than each other.

B. One tailed test: (Right tailed): First Sample mean is not more than second sample  mean. OR First Sample mean is equal to or less than second sample mean.

C. One tailed test: (Left tailed): First Sample mean is not less than second sample mean. OR First Sample mean is equal to or more than second sample mean.

 Alternate Hypothesis

A. Two tailed test: Sample means are significantly different than each other.

B. One tailed test: (Right tailed): First Sample mean is significantly more than second sample mean.

C. One tailed test: (Left tailed): First Sample mean is significantly less than second sample mean.

2. Calculate Test statistics Z

where t is the test statistics,
x̄ = Sample means for first or second sample, respective to its subscript.
SD = sample Standard deviations for first or second sample, respective to its subscript,
N = Sample Size in first or second sample, respective to its subscript.

3. Know p value from Z table

4. Interpret

If p < = alpha, then reject Null hypothesis, and accept alternate hypothesis.

If p > alpha, then the study has failed to reject Null hypothesis. So accept Null hypothesis.

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Confidence Intervals

1. For two tailed test:

100 - alpha % confidence interval = Actual difference in means ± critical_Z_value * SE_{Diff in means}

where critical_Z_value is the Z table value corrsponding to 2 tailed test and given alpha.

1. For one tailed test:

A. Right tailed test (AH = Sample mean is significantly more than hypothesized mean)

100 - alpha % confidence interval = Actual difference in means - critical_Z_value * SE _{Diff in means} to infinity.
where critical_Z_value is the t table value corrsponding to 1 tailed test and given alpha.

B. Left tailed test (AH = Sample mean is significantly less than hypothesized mean)

100 - alpha % confidence interval = minus Infinity to Actual difference in means + critical_Z_value * SE _{Diff in means}
where critical_Z_value is the Z table value corrsponding to 1 tailed test and given alpha.

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Example 1:

Random samples of 50 boys and 60 girls have shown mean IQ of 98 ± 6 and 100 ± 5, respectively. Apply an approriate statistical test to test whether there is gender-wise significantly difference in IQ, at alpha level of 5%.

As N > 30, appropriate statistical test will be "Z test."
Here,
x̄ ₁ = 98
SD₁ = 6
x̄ ₂ = 100
SD₁ = 5
N₁= 50
N₂= 60
alpha = 5%
tails = 2 (significantly different) Putting above values at the respective places, we get the output as follows: 

As p = 0.061, p is non-significant. We can say that there is no statistically significant difference between IQs in these two groups.

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@ Sachin Mumbare

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