# Z-Test (Z0, Ze & H0) for x̄1 = 78, x̄2 = 85, σ1 = 9, σ2 = 11, 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 two different population standard deviations σ1 = 9 & σ2 = 11, 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 σ19
Population Standard deviation σ211
Sample Size n165
Sample Size n275
Z04.1396

## Z-Test Work with Steps for x̄1=78 & x̄2 = 85 with unequal 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 two different population standard deviations σ1 = 9 & σ2 = 11 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 σ1 = 9
Population Standard deviation σ2 = 11
Sample Size n1 = 65
Sample Size n2 = 75
z score value (z) = 0.98
Formula
Z0=1 - x̄2
σ12/n1 + σ22/n2

step 2 Substitute x̄1, x̄2, σ1, σ2, n1 & n2 values in the formula

=78 - 85
((9)2 / 65) + ((11)2 / 75)

step 3 Simplify the above expression
=7
((9)2 / 65) + ((11)2 / 75)

=7
(81/ 65) + (121 / 75)

=7
1.2461538461538 + 1.6133333333333

=7
2.8594871794872

=71.691
Z0= 4.1396

Inference
There is significance difference, since the calculated value of Z0 = 4.1396 is greater than the table value of Ze = 0.98. Therefore the null hypothesis H0 is rejected. 