Z Test

• Assumptions
  

1. One sample Z test is used when the sample size is large (> 30) or population standard deviation is known.
   
2. Method of simple random sampling is used
   
3. Data is normally distributed
   

• 5-Step Hypothesis
  

Null and Alternative Hypothesis

• Ho: population means is equal to μo. μ = μo
• Ha1: population means is equal to μo. μ≠μo  (Two-tailed test)

Ha2: population means is equal to μo. μ<μo  (Left tailed test)

Ha3: population means is equal to μo. μ>μo  (Right tailed test)

Here μ_o takes a particular value.

Test Statistic

Critical value

• -z(a/2), z(a/2) (Two-tailed test)
• -z(a) (Left tailed test)
• z(a) (Right tailed test)

P-value

• 2*(1-P(Z≤|z|)  (Two-tailed test)
• P(Z≤z)  (Left tailed test)
• P(Z≥z)  (Right tailed test)

Decision rule

• Reject Ho if |z|>z(a/2) or p-value < alpha (two-tailed test)
• Reject Ho if –z < -z(a) or p-value < alpha (left tailed test)
• Reject Ho if z > z(a) or p-value < alpha (right-tailed test)

• Confidence Interval

Standard error (SE) and margin of error (ME) is given by:

100(1-alpha)% Confidence interval for the population mean is given by:

This implies I am 100(1-alpha)% confident that estimated population mean lies in the obtained interval.

One sample z test CALCULATOR