# Z-Test (Z_{0}, Z_{e} & H_{0}) for x̄_{1} = 78, x̄_{2} = 85, σ_{1} = 9, σ_{2} = 11, 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 two different *population standard deviation*s σ_{1} = 9 & σ_{2} = 11, 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_{1} | 9 |

Population Standard deviation σ_{2} | 11 |

Sample Size n_{1} | 65 |

Sample Size n_{2} | 75 |

Z_{0} | 4.1396 |

## Z-Test Work with Steps for x̄_{1}=78 & x̄_{2} = 85 with unequal 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 two different population *standard deviations* σ_{1} = 9 & σ_{2} = 11 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

_{1}= 9

Population Standard deviation σ

_{2}= 11

Sample Size n

_{1}= 65

Sample Size n

_{2}= 75

z score value (z) = 0.98

__Formula__

Z

_{0}=x̄

_{1}- x̄

_{2}

√

_{1}

^{2}/n

_{1}+ σ

_{2}

^{2}/n

_{2}

step 2 Substitute x̄

_{1}, x̄

_{2}, σ

_{1}, σ

_{2}, n

_{1}& n

_{2}values in the formula

=78 - 85

√

^{2}/ 65) + ((11)

^{2}/ 75)

step 3 Simplify the above expression

=7

√

^{2}/ 65) + ((11)

^{2}/ 75)

=7

√

=7

√

=7

√

=71.691

Z

_{0}= 4.1396

__Inference__There is significance difference, since the calculated value of Z

_{0}= 4.1396 is greater than the table value of Z

_{e}= 0.98. Therefore the null hypothesis H

_{0}is rejected.