# Z-Test (Z0, Ze & H0) for x̄1 = 78, x̄2 = 85, σ = 9, n1 = 65 & n2 = 75

Work with steps, formula & calculation summary to estimate the Z-statistic (Z0), critical (table) value (Ze) for degrees of freedom & hypothesis test (H0) at a stated level of significance for difference between two sample means1 = 78 & x̄2 = 85, with common population standard deviation σ = 9, and sample size n1 = 65 & n2 = 75. The below is the calculation summary for Z-statistic for the test of significance or hypothesis for difference between two sample means x̄1 = 78 & x̄2 = 85

Calculation Summary
Sample mean x̄178
Sample mean x̄285
Population Standard deviation σ9
Sample Size n165
Sample Size n275
Z04.5896

## Z-Test Work with Steps for x̄1 = 78 & x̄2 = 85 with common SD

The below is the work with step by step calculation shows how to estimate the Z-statistic (Z0), critical (table) value (Ze) for degrees of freedom & hypothesis test (H0) at a stated level of significance for difference between two sample means x̄1 = 78 & x̄2 = 85 with common population standard deviations σ = 9, and sample size n1 = 65 & n2 = 75 may help grade school students to solve the similar Z-test statistic (Z0) worksheet problems efficiently.

Workout :
step 1 Address the formula, input parameters and values
Sample mean x̄1 = 78
Sample mean x̄2 = 85
Population Standard deviation σ = 9
Sample Size n1 = 65
Sample Size n2 = 75
z score value (z) = 0.98
Formula
Z0=1 - x̄2
σ
1/n1 + 1/n2

step 2 Substitute x̄1, x̄2, σ, n1 & n2 values in the formula

=78 - 85
9
(1 / 65) + (1 / 75)

step 3 Simplify the above expression
=7
9
(1 / 65) + (1 / 75)

=7
9
0.015384615384615 + 0.013333333333333

=7
9
0.028717948717949

=79 x 0.1695

=71.5252

Z0 = 4.5896

Inference
There is significance difference
since the calculated value of Z0 = 4.5896 is greater than the table value of Ze = 0.98.
Therefore the null hypothesis H0 is rejected.