# Z-Test (Z_{0}, Z_{e} & H_{0}) for x̄_{1} = 78, x̄_{2} = 85, σ = 9, n_{1} = 65 & n_{2} = 75

Work with steps, formula & calculation summary to estimate the Z-statistic (Z_{0}), critical (table) value (Z_{e}) for degrees of freedom & hypothesis test (H_{0}) at a stated level of significance for difference between two sample *means* x̄_{1} = 78 & x̄_{2} = 85, with common *population standard deviation* σ = 9, and *sample size* n_{1} = 65 & n_{2} = 75.
The below is the calculation summary for Z-statistic for the test of significance or hypothesis for difference between two sample means x̄_{1} = 78 & x̄_{2} = 85

Calculation Summary | |
---|---|

Sample mean x̄_{1} | 78 |

Sample mean x̄_{2} | 85 |

Population Standard deviation σ | 9 |

Sample Size n_{1} | 65 |

Sample Size n_{2} | 75 |

Z_{0} | 4.5896 |

## Z-Test Work with Steps for x̄_{1} = 78 & x̄_{2} = 85 with common SD

The below is the work with step by step calculation shows how to estimate the Z-statistic (Z_{0}), critical (table) value (Z_{e}) for degrees of freedom & hypothesis test (H_{0}) at a stated level of significance for difference between two sample means x̄_{1} = 78 & x̄_{2} = 85 with common population *standard deviations* σ = 9, and sample size n_{1} = 65 & n_{2} = 75 may help grade school students to solve the similar Z-test statistic (Z_{0}) worksheet problems efficiently.

__Workout :__

step 1 Address the formula, input parameters and values

Sample mean x̄

_{1}= 78

Sample mean x̄

_{2}= 85

Population Standard deviation σ = 9

Sample Size n

_{1}= 65

Sample Size n

_{2}= 75

z score value (z) = 0.98

__Formula__

Z

_{0}=x̄

_{1}- x̄

_{2}

σ√

_{1}+ 1/n

_{2}

step 2 Substitute x̄

_{1}, x̄

_{2}, σ, n

_{1}& n

_{2}values in the formula

=78 - 85

9 √

step 3 Simplify the above expression

=7

9 √

=7

9 √

=7

9 √

=79 x 0.1695

=71.5252

Z

_{0}= 4.5896

__Inference__There is significance difference

since the calculated value of Z

_{0}= 4.5896 is greater than the table value of Z

_{e}= 0.98.

Therefore the null hypothesis H

_{0}is rejected.